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SCHAUM'S 




MATHEMATICAL 
HANDBOOK 

off FORMULAS and TABLES 



by MURRAY R. SPIEGEL 



including 

fformulcis 
I^O tables 



SCHAUM'S OUTLINE SERIES 

McGRAW-HILL BOOK COMPANY 



^n^m 



SCHAUM'S OVTLmE SERIES 



MATHEMATICAL 
HANDBOOK 



of 



Formulas and Tables 



BY 



MURRAY R. SPIEGEL, Ph.D. 

Professor of Mathematics 
Rensselaer Polytechnic Institute 



SCHAUM'iS OIJTLIAE SERIES 

McGRAW-HILL BOOK COMPANY 

New York, St. Louis, ^um^rancis£a»Soronlo, Sydney 




.- 'f ■ ^ 



Copyright © 1968 by McGraw-Hill. Inc. All Rights Reserved. Printed in the 
United States of America. No part of this publication may be reproduced, 
stored in a retrieval system, or transmitted, in any form or by any means, 
electronic, mechanical, photocopying, recording, or otherwise, without the 
prior written permission of the publisher. 



60224 
234567890 MHUN 72 10 698 



Tvpoaravhv by Signs and Symbola. Inc., New York, N. Y. 



Preface 

The purpose of this handbook is to supply a collection of mathematical formulas and 
tables which will prove to be valuable to students and research workers in the fields of 
mathematics, physics, engineering and other sciences. To accomplish this, care has been 
taken to mclude those formulas and tables which are most likely to be needed in practice 
rather than highly specialized results which are rarely used. Every effort has been made 
to present results concisely as well as precisely so that they may be referred to with a maxi- 
mum of ease as well as confidence. 

Topics covered range from elementary to advanced. Elementary topics include those 
from algebra, geometry, trigonometry, analytic geometry and calculus. Advanced topics 
mclude those from differential equations, vector analysis, Fourier series, gamma and beta 
functions, Bessel and Legendre functions, Fourier and Laplace transforms, elliptic functions 
and various other special functions of importance. This wide coverage of topics has been 
adopted so as to provide within a single volume most of the important mathematical results 
needed by the student or research worker regardless of his particular field of interest or 
level of attainment. 

The book is divided into two main parts. Part I presents mathematical formulas 
together with other material, such as definitions, theorems, graphs, diagrams, etc., essential 
for proper understanding and application of the formulas. Included in this first part are 
extensive tables of integrals and Laplace transforms which should be extremely useful to 
the student and research worker. Part II presents numerical tables such as the values of 
elementary functions (trigonometric, logarithmic, exponential, hyperbolic, etc.) as well as 
advanced functions (Bessel, Legendre, elliptic, etc.). In order to eliminate confusion 
especially to the beginner in mathematics, the numerical tables for each function are sep- 
arated. Thus, for example, the sine and cosine functions for angles in degrees and minutes 
are given in separate tables rather than in one table so that there is no need to be concerned 
about the possibility of error due to looking in the wrong column or row. 

I wish to thank the various authors and publishers who gave me permission to adapt 
data from their books for use in several tables of this handbook. Appropriate references 
to such sources are given next to the corresponding tables. In particular I am indebted to 
the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates F R S 
and to Oliver and Boyd Ltd., Edinburgh, for permission to use data from Table III of their 
book Statistical Tables for Biological, AgHcultural and Medical Research. 

I also wish to express my gratitude to Nicola Monti, Henry Hayden and Jack Margolin 
for their excellent editorial cooperation. 

M. R. Spiegel 
Rensselaer Polytechnic Institute 
September, 1968 



CONTENTS 




Page 

1 . Special Constants 1 

2. Special Products and Factors 2 

3. The Binomial Formula and Binomial Coefficients 3 

4. Geometric Formulas 5 

5. Trigonometric Functions ^1 

6. Complex Numbers 21 

7. Exponential and Logarithmic Functions 23 

8. Hyperbolic Functions 26 

9. Solutions of Algebraic Equations 32 

10. Formulas from Plane Analytic Geometry 34 

11. Special Plane Curves 40 

12. Formulas from Solid Analytic Geometry 46 

13. Derivatives ^ 

14. Indefinite Integrals 57 

15. Definite Integrals ^ 

16. The Gamma Function 101 

17. The Beta Function 103 

1 8. Basic Differential Equations and Solutions 104 

19. Series of Constants 107 

20. Taylor Series HO 

21 . Bernoulli and Kuler Numbers 114 

22. Formulas from Vector Analysis H® 

23. Fourier Series 131 

24. Bessel Functions 13^ 

25. Legendre Functions 14® 

26. Associated Legendre Functions 149 

27. Hermite Polynomials 1^1 

28. Laguerre Polynomials 1^ 

29. Associated Laguerre Polynomials 1^5 

30. Chebyshev Polynomials 1^*^ 



CONTENTS 

Page 

31. Hypergeometric Functions 160 

32. Laplace Transforms 161 

33. Fourier Transforms 174 

34. Elliptic Functions 179 

35. Miscellaneous Special Functions 183 

36. Inequalities 185 

37. Partial Fraction Expansions 187 

38. Infinite Products 188 

39. Probability Distributions 189 

40. Special Moments of Inertia 190 

41. Conversion Factors 192 




Sample problems illustrating use of the tables 194 

1. Four Place Common Logarithms 202 

2. Four Place Common Antilogarithms 204 

3. Sin X (x in degrees and minutes) 206 

4. Cos X {x in degrees and minutes) 207 

5. Tan x (x in degrees and minutes) 208 

6. Cotx {x in degrees and minutes) 209 

7. Sec X {x in degrees and minutes) 210 

8. Csc X {x in degrees and minutes) 211 

9. Natural Trigonometric Functions (in radians) 212 

10. log sin X {x in degrees and minutes) 216 

11. log cos X {x in degrees and minutes) 218 

12. log tan X {x in degrees and minutes) 220 

13. Conversion of radians to degrees, minutes and seconds 

or fractions of a degree 222 

14. Conversion of degrees, minutes and seconds to radians 223 

15. Natural or Napierian Logarithms lege a; or In a; 224 

16. Exponential functions c* 226 

17. Exponential functions e"'' 227 

18a. Hyperbolic functions sinh x 228 

18b. Hyperbolic functions cosh x 230 

18e. Hyperbolic functions tanhx 232 



CONTENTS 

Page 

19. Factorial n 234 

20. Gamma Function 234 

21. Binomial Coefficients 236 

22. Squares, Cubes, Roots and Reciprocals 238 

23. Compound Amount: (1 + r)" 240 

24. Present Value of an Amount: (1 + r)"" 241 

25. Amount of an Annuity: (^+^)"'~ ^ 242 

r 



26. Present Value of an Annuity: ^^'^'^^ 



r 243 

27. Bessel functions Jo{x) 244 

28. Bessel functions Ji{x) 244 

29. Bessel functions Yq{x) 245 

30. Bessel functions Yi{x) 245 

31. Bessel functions Iq{x) 246 

32. Bessel functions Ii{x) 246 

33. Bessel functions Ko{x) 247 

34. Bessel functions Ki{x) 247 

35. Bessel functions Ber (x) 248 

36. Bessel functions Bei (x) 248 

37. Bessel functions Ker (x) 249 

38. Bessel functions Kei (x) 249 

39. Values for Approximate Zeros of Bessel Functions 250 

40. Exponential, Sine and Cosine Integrals 251 

41. Legendre Polynomials Pn{x) 252 

42. Legendre Polynomials Pr.{cos 6) 253 

43. Complete Elliptic Integrals of First and Second Kinds 254 

44. Incomplete Elliptic Integral of the First Kind 255 

45. Incomplete Elliptic Integral of the Second Kind 255 

46. Ordinates of the Standard Normal Curve 256 

47. Areas under the Standard Normal Curve 257 

48. Percentile Values for Student's t Distribution 258 

49. Percentile Values for the Chi Square Distribution 259 

50. 95th Percentile Values for the F Distribution 260 

51. 99th Percentile Values for the F Distribution 261 

52. Random Numbers 262 

Index of Special Symbols and Notations 263 

Index 265 



Part I 



FORMULAS 



THE GREEK ALPHABET 



1 
Greek 
name 


Greek letter 


Lower case 


Capital 


Alpha 


CC 


A 


Beta 


/3 


B 


Gamma 


y 


r 


Delta 


s 


A 


Epsilon 


£ 


E 


Zeta 


t 


Z 


Eta 


V 


H 


Theta 


9 





Iota 


I 


I 


Kappa 


K 


K 


Lambda 


A 


A 


Mu 


t^ 


M 



Greek 
name 


Greek letter 


Lower case 


Capit£.l 


Nu 


V 


N 


Xi 


1 


H 


Omicron 








Pi 


TT 


n 


Rho 


P 


P 


SigTTia 


a 


2 


Tau 


T 


T 


Upsilon 


V 


Y 


Phi 


* 


* 


Chi 


X 


X 


Psi 


^ 


* 


Omega 


u 


a 



1.1 TT = 3.14159 26535 89793 23846 2643... 

1.2 e = 2.71828 18284 59045 23536 0287... = lim f 1 + -)" 

= natural base of logarithms «-» \ n/ 

1.3 V2 =: 1.4142135623 73095 0488... 

1.4 ^/3 := 1.73205 08075 68877 2935... 
IS v^ ^ 2.23606 79774 99789 6964... 

3 _ 

1.6 V2 = 1.25992 1050... 

1.7 ^/S = 1.44224 9570... 

1.8 y/2 = 1.14869 8355... 

1.9 ^ = 1.24573 0940... 

1.10 e"" = 23.14069 26327 79269 006... 

1.11 n-e = 22.45915 77183 61045 47342 715... 

1.12 e^ = 15.15426 22414 79264 190... 

1.13 log,o2 = 0.30102 99956 639811952137389... 

1.14 logioS ^ 0.47712 12547 19662 43729 50279... 
1-15 logio e = 0.43429 44819 0325182765... 

1.16 log,o,r = 0.49714 98726 94133 85435 12683... 

1.17 log^ 10 ^ InlO = 2.30258 50929 94045 684017991... 

1.18 ]og^2 = ln2 = 0.69314 71805 59946 309417232... 

1.19 log, 3 = In 3 = 1.0986122886 68109 69139 5245... 
y = 0.5772156649 01532 86060 6512... = Euler's constant 



1.23 

1.24 
1.25 



1.20 



1.21 ey ^ 1.78107 24179 90197 9852... [see 1.20] 

1.22 ^fe = 1.64872 12707 00128 1468... 

V^ = r(^) = 1.77245 38509 05516 02729 8167... 
where r is the gamma function [see pages 101-1021. 

r(^) = 2.67893 85347 07748... 

r(^} = 3.62560 99082 21908... 

1.26 1 radian = l%0°h ~ 57.29577 95130 8232...° 

1.27 1° ^ ,r/180 radians ^ 0.01745 32925 19943 2957. .. radians 



2.1 ix + y)^ = x2 + 2xy + y^ 

2.2 {X - j/)2 = x2 - 2xy + y^ 

2.3 (a: + j/)3 = x3 + 3x2]/ + Sxy2 + j,3 

2.4 (x - ^)3 = x3 - 3a:2i/ + 3x1/2 - j/3 

2.5 (X + y)* = X* + iX^y + 6x2j/2 + 4a;y3 + j^ 

2.6 (» - y)-* = X*- Ax^y + 6x2]/2 - 4xy3 + y^ 

2.7 (X + y)5 = x5 + 5r»j/ + 10x3i/2 + 10x2y3 + ^y;yi + j^5 

2.8 {x - j/)5 = x5 - 5x^1/ + 10x3^2 _ I0x2y3 + 5a:y4 _ yS 

2.9 (x + y)^ = x« + 6x5i/ + 15x4i/2 + 20xV + I5a:2y4 + gxyS + y6 

2.10 (x - i/)fl = x8 - 6x5y + 15»*i/2 - 20x3tf3 + 15x2y4 - 6x^/5 + y^ 

The results 2.1 to 2.10 above are special cases of the binomial formula [see page 3]. 

2.11 a:2- y2 = (a; - y)(a; + y) 

2.12 x3-|/3 = (X - 1/)(X2 + xy + 1/2) 

2.13 X3 + y3 = (;;e + y)(a;2_a;y + y2) 

2.14 z*~y* = (a: - y)(x + y)(x2 4- y2) 

2.15 x5 - yS = (x-y)(x4 + a;3y + x2y2 + a:y3 + y4) 

2.16 x5 + y5 = (x + y)(x* — x3y + x2y2 - xy3 + y*) 

2.17 x6-y6 = {x-y)ix + y)(x^ + xy-hy^){x^-xy + yi) 

2.18 X4 + X2y2 + y4 = (a;2 + :cy + y2)(a;2 _ a;y + y2) 

2.19 x* + 43/4 = (a.2 + 2xy+2y2)(x2-2xj/ + 2y2) 

Some generalizations of the above are given by the following results where m is a positive integer. 



2.20 x2" + l - y2n + i = (a- - y)(a;2" + x2n-ly + x2"-2y2 + . - . + y2n) 



2xy cos 



= (x - y) X 



x2 — 2xy cos 



2n + 1 
2nir 



2n+l 



+ y2 )( sc2 — 2xy cos 



+ y2 



Air 



2n + l 



+ y2 



2.21 x2"+l + y2n+l = (x + y)(x2n - x2''-ly + x2n-2y2 _ . . . + y2n) 

2 



= (x + y) { x2 + 2xy cos^ — x~r + y^ )( a^ + 2xy cos _ 
x2 + 2xy cos ^ — -^ + y2 



^.vO 



2.22 



r2n _ ,,2n — 



— (* ~ y)(3^ + y)(a:"~' + x'*~2y + 3.n-3y2 + . . .)(a.n-i _ x^^^y + a;"~3y:i _ . . .) 

— (sJ ~ J/){^ 4- y) ( x2 — 2xy cos— + y2 )[ x2 — 2xy eos-^ + y^ 
■ ~ 2xy cos ^" ~ ^^"' + y2 



2^3 



+ y2n - ( x2 + 2xy cos^ + y2 J( a;2 + 2xy cos |^ + y 



2n 



2n 



•} I n (2n — l)jr , n 

x'- + 2a:y cos -z + y2 



3 



The BINOMIAL FORMULA 
and BINOMIAL COEFFICIENTS 



If 71 — 1,2,3, . . . factorial n or n factorial is defined as 
3.1 n! = 1-2-3 n 



We also define zero factorial as 



3.2 



0! ^ 1 



mwf^m fm^im mB^ 



If 71 = 1,2,3, ... then 
3.3 {x + vr = a;" + nx^^^y + ^^^^^f^ a:"-2y2 + "^"^ ~ g^"" ^^ a:"^3y3 + ... + y« 

This is called the binomta? formula. It can be extended to other values of n and then is an infinite series 
[see Binomial Series, page 110]. 




BINOMIAL COEFFICIENTS 

imiiiiiiiiiiiiiiii^ 



The reaxilt 3.3 can also be written 



3.4 (x + j/)" = x^ + ('^Jx'^-^y + f 2Jx"-2y2 + [^ )x^-3y^ + 

where the coeflicients, called binomial coefficients, are given by 



3.5 



n\ ^ n{n - l){n - 2)- • -{n- k + 1) ^ n\ 

kj k\ k\(n-k)\ 



'n 



n 
n ~ k 



4 



THE BINOMIAL FORMULA AND BINOMIAL COEFFICIENTS 



PROPERTIES OF BINOMIAL COEFFICIENTS 



3.6 



+ 



« + 1 

k-{- 1 



k / V A; + 1 
This leads to Paacal's triangle [see page 236]. 



3.7 



n\ in 

+ 



ly + UJ + 



+ 



= 2" 



3.8 



(;)-(:)-(;)-•■•<-(:) 



= 



3.9 



C) 



■nl \ n I \ n I \ n 



« + »i + 1 
n + 1 



3.10 



C) - (: 



+ (4 ) + 



= %v.-\ 



3.11 



vm:^-^"'- 



= 2"-* 



3.12 



+ 



+ 



+ ■■■ + 



3.13 



:)(:)-(:)(.-.- 






m + m 
P 



3.14 



/n 



(Di^lJ + (2)1^2; + (3}(^3J + ■ 



+ («1 



■= n2"-' 



/?i 



3.15 (l)(^i^ - (2)(^2y + (3)(^3 



,(„!)« + !(„)( ^ ) = 



MULTINOMIAL FORMULA 



3.16 



(ici + a:^ + ■ • ■ + iCp)" = 5 



w! _n, „n« _n, 






where the sum, denoted by 5, is taken over all nonnegative integers n^tU-i, 



. , n^ for which 




GEOMETRIC FORMULAS 



RECTANGLE OF LENGTH 6 AND WIDTH o 



4.1 Area = ab 

4.2 Perimeter = 2a + 2b 



b 
Fig. 4-1 



PARALLELOGRAM OF ALTITUDE h AND BASE h 



4.3 Area = bk = ab sin e 

4.4 Perimeter = 2a + 26 




b 

Fig. 4-2 




4.5 



RIANGLE OF ALTITUDE h AND BASE 5 



Area = ^hh = ^ab sin e 

where s^ ^{a + b + c) = semiperimeter 
4.6 Perimeter = a + b + c 




4.7 

4.8 



TRAPEZOID OP ALTITUDE h AND PARALLEL SIDES a A 



N 



Area - ^k{a + b) 

Perimeter = a + b -i- h ( — L -\ ^ 

\sinff sin^ 

= a + b + /i(csc e + CSC -p) 




b 
Pig. 4-4 




GEOMETRIC FORMULAS 



4.9 
4.10 



4.15 



REGULAR POLYGON OF n SIDES EACH OF LENGTH b 



Area = ^nb^ cot— = Inb^^ 



cos JTr/n) 
sin (ir/n) 



Perimeter = nb 



CIRCLE OF RADIUS r 



4.11 Area = trr^ 

4.12 Perimeter = 2irr 



SECTOR OF CIRCLE OF RADIUS r 



4.13 Area - ]^r^$ [ff in radians] 

4.14 Arc length 8 = re 



RADIUS OF CIRCLE INSCRIBED IN A TRIANG 



\/8(« — a)(s — 6)(« — c) 



where s = ^(a + b + c) — semiperimeter 




Fig. 4-5 




Fig. 4-6 



RADIUS OF CIRCLE CIRCUMSCRIBING A TRIANGLE OF SIDES a,b,c 




4.16 



R = 



abc 



4^/s{8 — a)(s — 6)(s — c) 
where s = ^(a+ b + e) — semiperimeter 




Fig. 4-9 



GEOMETRIC FORMULAS 



REGULAR POLYGON OF n SIDES INSCRIBED IN CIRCLE OF RADIUS r 



4.17 Area = ^nr^ sin ~ = ^nr^ sin ^^ 

4.18 Perimeter = 2nrsin— = 2nr sin 

n ft 



n SIDES CIRCUi 



REGULAR POLYG 



4.19 Area = nr^ tan- = nr^ t&n^^ 

n n 

4.20 Perimeter = 2nr tan — = 2nr tan ^^ 

71 n 



IE6MENT OF CIRCLE OF RADIUS r 



4.21 Area of shaded part = ^+-2 {$ — sin tf) 




Fig. 4-11 




Fig. 4-12 



ELLIPSE 



nkmmmMMimkimmm^ 



4.22 Area = vob 

r/2 
y/l — 



k^ sin2 $ de 



— 2v y/^{a^ + b^) [approximately] 
where k = Va^ — by a. See page 254 for numerical tables. 



fMililiJB^ AXIS b 




Fig. 4-13 



lEGMENT OF A PARABO 



4.24 Area = %ab 



, 1,2 /4a + y/b^'+lG^ 

4.25 Arc length ABC = A VP+Jg^ + ~ In 




Fig. 4-14 



8 



GEOMETRIC FORMULAS 



RECTANGULAR PARALLELEPIPED OF LENGTH a, HEIGHT I, WIDTH 



4.26 Volume = ahc 

4.27 Surface area = 2(o6 + tie + 6c) 







a 
Fig. 4-15 



PARALLELEPIPED OF CROSS-SECTIONAL AREA ^ AND HEIGHT h 



4.28 Volume = Ah — ahc si 



sin e 




Fig. 4-16 



SPHERE OF RADIUS r 



4.29 Volume := ^.^ 

4.30 Surface area = Attt'^ 




Fig. 4-17 
RIGHT CIRCULAR CYLINDER OF RADIUS r AND HEIGHT h 



4.31 Volume = irr^h 

4.32 Lateral surface area = 'i.Trrh 



Fig. 4-18 
CIRCULAR CYLINDER OF RADIUS r AND SLANT HEIGHT I 

4.33 Volume ~ tttH = —. — ■ = trr'^h esc B 

sin e 

4.34 Lateral surface area = %^tI — ~- — = 2?rr^ esc B 

sin B 

Fig. 4-19 




GEOMETRIC FORMULAS 

CYLINDER OF CROSS-SECTIONAl AREA A AND SLANT HEIGHT I 

4.35 Volume = Al = — — = Ah esc e 

sine 

4.36 Lateral surface area = vl = -?■ — ■= ■ph cse B 

sme 



Note that formulas 4.31 to 4.34 are special cases. 




Fig. 4-20 



RIGHT CIRCULAR CONE OF RADIUS r AND HEIGHT h 



4.37 Volume = lir-r^h 



4.38 Lateral surface area - vr v'r2"4~P = wrl 




Fig. 4-21 



PYRAMID OF BASE AREA A AND HEIGHT h 



4.39 Volume = ^Ah 




SPHERICAL CAP OF RADIUS r AND HEIGHT h 



Fig. 4-22 



4.40 Volume {shaded in figure) = ^7r/t2{3r - h) 

4.41 Surface area = 2irrh 



Fig. 4-23 
FRUSTRUM OF RiGHT CIRCULAR CONE OF RADII a.b AND HEIGHT h 




4.42 Volume ^ ^^h{a^ -\- ah -\- h^) 



4.43 Lateral surface area = Tr{a + h) yjh^ + (6 - a)2 




Fig. 4-24 



10 



GEOMETRIC FORMULAS 



PHERICAL TRIANGLE OF ANGLES A,B,C ON SPHERE OF RADIUS r 



4.44 Area of triangle ABC = {A + B + C - jr)r'^ 




Fig. 4-25 



TORUS OF INNER RADIUS a AND OUTER RADIUS b 



4.45 Volume - iTrHa + b)(b - a)2 

4.46 Surface area = Jr2(b2 - ^2) 




Pig. 4-26 



;OID OF SEMI-AXES a.b^c 



4.47 Volume — jvabc 




Fig. 4-27 



PARABOLOID OF REVOLUTION 



4.48 Volume = j^irb^a 




Fig. 4-28 



DEFINITION O F TRIGONOMETRIC FUNCTIONS FOR A RIGHT TRIANGLE 



J 



Triangle ABC has a right angle (90°) at C and sides of length a, 6, c. The trigonometric functions of 
angle A are defined as follows. 



5.1 
S.2 
5.3 
5.4 
BS 
5.6 



. . a, opposite 

»i7te of A = sin A = - = .„„„■„„.. — 
c hypotenuse 

. . A ^ adjacent 

coaiTie ot A = cos A = — — 



c hypotenuse 

, . ^ . O' opposite 

tangent ot A - tan A - ^ - ^j^^^ 



cotangent of A = cot A 



& _ adjacent 
a opposite 



, . . c hypotenuse 

secant of A = sec A = -r = — j-- t— 

b adjacent 

_. . . A ^ hypotenuse 

cosecant of A = esc A = — = - — ~ — tt — 

a opposite 




Fig. 5-1 



EXTENSIONS TO ANGLES WHICH MAY BE GREATER THAN 90' 



Consider an xy coordinate system [see Fig. 5-2 and 5-3 below]. A point P in the xy plane has coordinates 
{x,y) where x is considered as positive along OX and negative along OX' while y is positive along py and 
negative along OY'. The distance from origin O to point P is positive and denoted by r — y/x^ + y^ . 
The angle A described counterclockwise from OX is considered positive. If it is described clockwise from 
OX it is considered negative. We call X'OX and Y'OY the x and y axis respectively. 

The various quadrants are denoted by I, II, III and IV called the first, second, third and fourth quad- 
rants respectively. In Fig. 5-2, for example, angle A is in the second quadrant while in Fig. 5-3 angle A 
is in the third quadrant. 





11 



12 



TRIGONOMETRIC FUNCTIONS 



For an angle A in any quadrant the trigonometric functions of A are defined as follows. 
5-/ sin A = y/r 



5.8 

5.9 

5.10 

5.11 

5.12 



cos A = xfr 

tan A = y/x 

cot A = x/y 

sec A = r/x 

CSC A = r/y 



RELATIONSHIP BETWEEN DEGREES AND RADIANS 



A radian is that angle 6 subtended at center O of a circle by an arc 
MN equal to the radius r. 

Since 27r radians = 360° we have 

5.13 1 radian ^ 180°/;r = 57.f9577 95130 8232. .. ° 

5.14 1° = B-/180 radians ^ 0.1745 32925 19943 2957. . .radians 




Fig. 5-4 



5.15 



5.18 



RELATIONSHIPS AMONG TRIGONOMETRIC FUNCTIONS 



tan A 



sin A 



5.16 cotA = 



5.17 sec A = 



CSC A = — : 



cos A 




1 
tan A 


cos A 

sin A 


1 

cos A 




1 





sin A 



5.19 sin2A + cos^A = 1 

5.20 sec2A - Un2A = 1 

5.21 csc2A - cot2A - 1 



"AMErVARIATIONS OF TRIGONOMETRIC FUNCTIONS 



Quadrant 


sin A 


cos A 


tan A 


cot A 


sec A 


esc A 


I 


-T 

to 1 


+ 
1 to 


+ 

to so 


+ 
=° to 


+ 
1 to «; 


+ 

00 to 1 


II 


+ 
1 to 


to -1 


~« to 


to -« 


-K to -1 


+ 
1 to =o 


III 


to -1 


-1 too 


to ~ 


-t- 
« to 


-1 to -» 


— re to —1 


IV 


-I to 


+ 
Otol 


-« to 


to -«> 


+ 

== to 1 


— 1 to —00 



TRIGONOMETRIC FUNCTIONS 



13 



EXACT VALUES FOR TRIGONOMETRIC FUNCTIONS OF VARIOUS ANGLES 



Angle A 
in degrees 


Angle A 
in radians 


sin A 


cos A 


tan A 


cot A 


sec A 


CSC A 


0° 








1 





00 


1 


00 


15° 


7r/12 


liV6-V2) 


i(\/6 + V^) 


2-^/z 


2 + ^/3 


^/6-^/2 


Ve + \/2 


30° 


:r/6 


i 


iVs 


iV3 


V3 


fV3 


2 


45° 


;r/4 


IV2 


iV^ 


1 


1 


\/^ 


V2 


60° 


HZ 


iVs 


1 
2 


V3 


^\/3 


2 


iV§ 


75° 


5W12 


liV6 + V2) 


1(Vg-V2) 


2 + V^ 


2-\/3 


\/6 + \/2 


V6-\/2 


90° 


7r/2 


1 





±00 





±co 


1 


105° 


77r/12 


l{\/6 + \/2) 


-i(V6-V^) 


-(2 + -/3 ) 


-{2-V^) 


-(\/6 + V2) 


\/6- VZ 


120° 


2B-/3 


1^3 


-i 


-\/3 


-^Vs 


-2 


fv^ 


135° 


3W4 


iV2 


-iV2 


-1 


-1 


-^ 


\/^ 


150° 


57r/6 


1 


-iV3 


-*V3 


-v^ 


-fv^ 


2 


165° 


lln-/12 


1(^6 -V^) 


-i(v/6 + \/2) 


H2-\/3) 


-(2 + V3) 


-(\/6-V2) 


V6 + \/2 


180° 


IT 





-1 





^oo 


-1 


±00 


195° 


13W12 


~1(Vg-V2) 


-i(V6 + \/2) 


2- \/3 


2 + \/3 


-(V6-\/2) 


-i^/G + ^/2) 


210° 


lir/e 


-i 


-1^3 


4V3 


V3 


-|V3 


-2 


225° 


5)r/4 


-^V2 


-i\/2 


1 


1 


~V2 


-V2 


240° 


47r/3 


-iVa 


—1 
2 


v/3 


i\/3 


-2 


-fV3 


255° 


1777-/12 


-i(V6 + v^) 


-HVe~V2) 


2 + \/3 


2 - VS 


-(\/6 + \/2) 


-(V6-V2) 


270° 


37r/2 


-1 





±0= 





^:oo 


~1 


285° 


197r/12 


-i(V6 + \/2) 


i{V6--v/2) 


-(2 + \/3 ) 


~(2~^/Z) 


V^ + \/2 


-(\/6-\/2) 


300° 


577/3 


-iVs 


^ 


-^/3 


-iVs 


2 


-fV3 


315° 


77r/4 


-iV2 


4V2 


-1 


-1 


V^ 


-V2 


330° 


11W6 


-I 


iV^ 


-iv^ 


-v^ 


§V^ 


-2 


345° 


23^7/12 


-i(\/6-V2) 


i(-/6 + \/2) 


-(2-\/3) 


-(2 + VS ) 


■/6-\/2 


-(\/6 + V2) 


360° 


2ir 





1 





q:(C 


1 


+ «= 



For tables involving other angles see pages 206-211 and 212-215. 



»■ — MT" 



14 



TRIGONOMETRIC FUNCTIONS 



GRAPHS OF TRIGONOMETRIC FUNCTIONS 



In each graph x is in radians. 
5.22 y = sin a: 

y 




Fig. 5-5 



5.24 y = tanx 




Fig. 5-7 



5.26 



y = sec a; 
J/ 









Fig. 5-9 



5.23 



y — cos X 




Fig. 5-6 




Fig. 5-8 



5.27 



y = CSC X 




O 



Fig. 5-10 



FUNCTIONS OF NEGATIVE ANGLES 



2:r 




5.28 sin (—A) = —sin A 
5.31 CSC (—A) = — c3Ci4 



5.29 cos (— /4) = cosA 
5.32 sec (—A) = sec A 



5.30 tan(-A) 
5.33 cot(-A) 



— tan A 

— cot A 



TRIGONOMETRIC FUNCTIONS 



16 



ION formul: 



5.34 
5.35 

5.36 
5.37 



sin (A^B) = 

cos (A ± B) = 

tan (A±B) = 

cot (A^B) = 



sin A cos B ± cos A sin B 
cos A cos B m: sin A sin fi 

tan A ± tan g 

1 ^ tan A tan B 

cot A cot B ^ I 
cot A ± cotB 



FUNCTIONS OF ANGLES IN AIL QUADRANTS IN TERMS OF THOSE IN QUADRANT I 





-A 


90^ ± A 

- + A 
2 -^ 


180^ i A 

rr ± A 


270"= It A 

^±A 
2 


A:{360°) It A 

2fcs- ± A 
& — integer 


sin 


— sin A 


cos A 


51 sin A 


— cos A 


It sin A 


cos 


cos A 


^ sin A 


— cos A 


± sin A 


cos A 


tan 


— tan A 


+ cot A 


±tanA 


ip cot A 


-^ tan A 


CSC 


— CSC A 


sec A 


ip CSC A 


— sec A 


It CSC A 


sec 


sec A 


~ CSC A 


— sec A 


± CSC A 


sec A 


cot 


— cot A 


wi tan A 


ieotA 


ip tan A 


±cotA 



RELATIONSHIPS AMONG FUNCTIONS OF ANGLES IN QUADRANT I 





sin A — u 


cos A — M 


tan A = u 


cot A = ti 


sec A = u 


CSC A ~ u 


sin A 
cos A 
tan A 
cot A 
sec A 
CSC A 


u 










1/m 




w/Vl + "2 


1/Vl + 1*2 


Va2 - 1/m 
1/u 


Vi - ?(2 


l/Vl + «2 
1/u 


j(/Vi + «2 

1/m 

u 


\/u2 — l/z( 


«/Vl - w^ 


Vi - «Vzt 


y/v?-\ 


1/Vu2 - 1 


V 1 ~ u^/u 


m/Vi - «2 

1/m 


iNu"^ - 1 

M 


Vw^-l 


1/Vl - m2 

1/m 


Vi + it^ 


Vl + W^M 


m/Vm^ - 1 


1/Vl - W2 


Vl + uVu 


Vl + M^ 


k/Vm2 - 1 


Vl + «2 



For extensions to other quadrants use appropriate signs as given in the preceding table. 



16 



TRIGONOMETRIC FUNCTIONS 



5.38 
5.39 



5.40 



sin2A 
cos2A 

tan2A 



DOUBLE ANGLE FORMULAS 



2 sin A cos A 

cos^A — sin^A 

2 tan A 
1 - tan2 A 



= 1 - 2 sin2 A = 2 cos^ A 



r 



5.41 



5.42 



5.43 



. A 
sin- 



COSg 



* A 
tan- 



HALF ANGLE FORMULAS 



V 



1 — cos A 






1 + COS A 
2 

1 — cos A 
1 + cos A 

sin A 



+ if A/2 is in quadrant I or II 

— if A/2 is in quadrant III or IV 

+ if A/2 is in quadrant I or IV 

— if A/2 is in quadrant II or III 

+ if A/2 is in quadrant I or III 

— if A/2 is in quadrant II or IV 
1 — cos A 



1 + cos A 



sin A 



— esc A — cot A 



MULTIPLE ANGLE FORMULAS 



5.44 
5.45 

5.46 

5.47 
5.48 

5.49 

5.50 
5.51 

5.52 



sinSA 
cosSA 

tanSA 

sin4A 
cos4A 

tan 4A 

sin&A 
cosSA 

tan5A 



See also formulas 5.68 and 5.69. 



3 sin A — 4 sin^ A 

4 cos^ A — 3 cos A 

3 tan A - tan^A 
1-3 tan2A 

4 sin A cos A — 8 sin^ A cos A 
8 cos-* A - Scos^A + 1 

4 tan A — 4 tan'* A 
1-6 tan2 A + tan^ A 

5 sin A - 20 sin^ A + 16 sin^ A 
16 cos^ A — 20 cos^ A + 5 cos A 

tanS A - 10 tan^ A + 5 tan A 
1 - 10tan2 A + Stan^A 



POWERS OF TRIGONOMETRIC FUNCTIONS 



5.53 sin2A ^ ^-^cos2A 

5.54 cos2A = | + ^cos2A 

5.55 sin^A — £ sin A — ^ sinSA 

5.56 cos'^fA — £cosA + icosSA 
See also formulas 5.70 through 5.73. 



5.57 


sin^ A 


5.58 


cos^ A 


5.59 


sin^ A 


5.60 


cos^ A 



^ — ^ COS 2A + ^ cos 4A 
f + ^ cos 2A + ^ cos 4A 
f sin A — ^ sin 3A + ^ sin 6A 
; 3A + ^ cos 5A 



§ cos- 



^cos; 



TRIGONOMETRIC FUNCTIONS 



17 



M, DIFFERENCE AND PRODUCT OF TRIGONOMETRIC FUNCTIO:4S 

5.61 sin A + sin B - 2 sin ^(-4 + B) cos ^(A - B) 

5.62 sin A - sin B = 2 cos ^(A + F) sin ^{A - B) 

5.63 cos A + cos F = 2 cos ^(A + B) cos ^{A -^ B) 

5.64 cos A - cos B = 2 3in^(A + B) sin i(B - A) 

5.65 sin A sin B = ^{cos (A - B) - cos (A + B)} 

5.66 cos A cos B = |{cos (A - B) + cos (A + B)} 

5.67 sinAcosB = ^{sin (A - B) + sin (A + B)} 



GENERAL FORMULAS 



5.68 sinTiA ^ sinA ■j(2cosA)"-i - f " ^ ) (2 cos A)"-3 + f '^ 2 j (2 cos A)"-5 - ■ • ■ I 

5.69 cosnA ^ |-^(2cosA)" - y{2cosA)"-2 + |-("~^j(2cosA)''-4 



n In — 4 



3 V 2 



(2cosA)''-6 + 



5.70 sin^n-iA = 



5.71 cos2"-iA = 



<g^|sin(2n-l)A - (^"^ ^ ] sin (2n - 3)A + 



4)' 



^-^^-^^t'-l'^^^"^ 



^A_|eos(2n-l)A + (^"^ ^)cos(2n-3)A + ••• + (^^_ ^^) cosaI 



5.72 sin2nA = 



5.73 cos2"A = 



= 2^(1") +|2;r^|cos2nA -(2«)cos(2n-2)A + ■•- ("D-^ (^^^j )cos2A 
h^l) + 2^f<'«2nA + (2n^ cos (2«-2)A + 



+ ( ^ J* ^ ) cos 2A 



If a: — sin J/ then y — sin~' x, i.e. the angle ■whose sine is x or inverse sine of x, is a many-valued 
function of x which is a collection of single-valued functions called branches. Similarly the other inverse 
trigonometric functions are multiple-valued. 

For many purposes a particular branch is required. This is called the principal branch and the values 
for this branch are called principal values. 



18 



TRIGONOMETRIC FUNCTIONS 



NCiPAL VALUES FOR rNVERSE TRIGONOMETRIC FUNCTIO 



Principal values for a; ^ 


Principal values for a- < 


£ sin-' a; ^ W2 


"W2 S sin-' a; < 


^ cos-^iB ^ jr/2 


5r/2 < cos-la: ^ tt 


^ tan-la; < irll 


— W2 < tan-' a; < 


< cot-'a; g Tr/2 


7r/2 < cot-ix < TT 


^ see-' a: < jr/2 


W2 < sec-' a; ^ - 


< csc-'a: ^ jr/2 


-7r/2 ^ csc-'x < 



REt^TtON$ BETWEEN INVERSE TRIGONOMETRIC FUNCTIONS 



In all cases it is assumed tbat principal values are used. 

5.74 sin-' x + cos-' a: = 7r/2 

5.75 tan-'x + cot-'ar ^ 7r/2 

5.76 aec-'a: + esc-' a; = 57/2 

5.77 CSC"' X — sin-' (1/a:) 

5.78 sec-' 3; — cos-' (1/a;) 

5.79 cot-' X = tan-' (1/a;) 



5.80 


sin-' (— fl!) 


— 


— sin-' X 


5.81 


COS-' {—x) 


= 


TT — COS-' X 


5.82 


tan-M-*) 


= 


— tan-' X 


5.83 


cot-' (-a:) 


= 


TT — cot-' X 


5.84 


sec-' (— x) 


= 


v — sec- ' * 


5.85 


CSC"' (—x) 


= 


— CSC-' X 



INVERSE TRIGONOMETRIC 



In each graph y is in radians. Solid portions of curves correspond to principal values. 



5.86 y — sin-'x 



5.87 y ~ cos-' a; 



5.88 y = tan-'x 



S-/2- 



-W2 ■ 



Fig. 5-11 



O : 



V 


y 




^^ 


K 


X 


-I 


A 




/ 

/ 
/ 
1 






; -' 


■ 





y 


V 


7r/2 


f X 


_^ 





-H2 


-""^ 



Fig. 5-12 



Fig. 5-13 



TRIGONOMETRIC FUNCTIONS 



19 



5.89 y = cot-^a; 




Fig. 5-14 



5.90 y = sec~'x 

y 



t/2 



-1 

-ir/2 



O i\ 



5.91 




Fig. 5-15 



Fig. 5-16 



RELATIONSHIPS BETWEEN SIDES AND ANGLES OF A PLANE TRIANGLE 



The following results hold for any plane triangle ABC with 
sides a, b, c and angles A, B, C. 

5.92 Law of Sines 



5.95 



sin A sin B sin C 

5.93 Law of Cosines 

c2 = o2 + 62 _ 2a6 COS C 
with similar relations involving the other sides and angles. 

5.94 Law of Tangents , ,,^10* 

a + h _ tan IJA + B) 

a- b ~ tan ^(A - B) 
with similar relations involving the other sides and angles. 




sin A = T- y/a{8 — a)(8 — b){a — c) 



Fig. 5-17 



where s = |(a +b + c) is the semiperimeter of the triangle. Similar relations involving angles 
B and C can be obtained. 
See also formulas 4.5, page 5; 4.15 and 4.16, page 6. 



RELATIONSHIPS BETWEEN SIDES AND ANGLES OF A SPHERICAL TRIANGLE 



Spherical triangle ABC is on the surface of a sphere as shown 
in Fig. 5-18. Sides a,b,c [which are arcs of great circles] are 
measured by their angles subtended at center O of the sphere. A, B, C 
are the angles opposite sides a,b,c respectively. Then the following 
results hold. 



5.96 Law of Sines 



sin a sin b 



sin A BinB sinC 
5.97 Law of Cosines 

cos o = cos b cos c + sin 6 sin c cos A 
cos A = — cos B cos C + sin B sin C cos a 
with similar results involving other sides and angles. 



Fig. 5-18 



] 




20 



TRIGONOMETRIC FUNCTIONS 



5.98 Law of Tangents 

tan ^(A + B) _ tan ^(a + b) 
tan ^{A-B) ~ tan |(o - b) 
with similar results involving other sides and angles. 



5.99 cos— = ^I sin8sin(i 

2 V sin 6 sii 

where s = ^(a + 6 + c). Similar results hold for other sides and angles. 



5.100 



(s-c) 
sin c 



cos 



a _ j cosjS-B) cos (S-C) 
2 V sin 5 sin C 



where S ~ ^(A + B + C). Similar results hold for other sides and angles. 
See also formula 4.44, page 10. 



NAPIER^-S^ RULES FOR RIGHT ANGIED SPHERICAL TRIANGLES 



Except for right angle C, there are five parts of spherical triangle ABC which if arranged in the order 
as given in Fig. 5-19 would be a,b,A,c,B. 





co-B 



co-A 



co-c 



Fig. 5-19 



Fig. 5-20 



Suppose these quantities are arranged in a circle as in Pig. 5-20 where we attach the prefix co 
[indicating coviplement] to hypotenuse c and angles A and B. 

Any one of the parts of this circle is called a middle part, the two neighboring parts are called 
adjacent parts and the two remaining parts are called opposite parts. Then Napier's rules are 

5.101 The sine of any middle part equals the product of the tangents of the adjacent parts. 

5.102 The sine of any middle part equals the product of the cosines of the opposite parts. 
Example: Since co-A = 90° -A, co-B = 90°— B. we have 

sin a — tan b tan (co-B) or sin o — tan b cot B 

sin (co-A) = cos a cos (co-B) or cos A = cos a sin B 
These can of course be obtained also from the results 5.97 on page 19. 



DEFINITIONS INVOLVING COMPLEX NUMBER 



A complex number is generally written as a + bi where o and b are real numbers and i, called the 
imaginary unit, has the property that v^ - ~1. The real numbers a and 6 are called the real and imaginary 
parts oi a + bi respectively. 

The complex numbers a + bi and a — bi are called complex conjugates of each other. 



EQUALITY OF COMPLEX NUMBEI 



6.1 



a + bi — c + di if and only if a = c and b = d 




6.2 



{a+bii-i- ic + di) = {a+c) + ib + d)i 



SUBTRACTION OF COMPLEX NUMBERS 



6.3 



(a + bi) - (c + di) = (a~c) + {b-d)i 



MULTIPLICATION OF COMPLEX NUMBERS 



6.4 



(a+bi){c + di) = {ae - bd) + {ad + bc)i 



DIVISION OF COMPLEX NUMBE 



6.5 



a + bi _ a + bi ^ c — di _ ac + bd f be — a,d\ . 
c + di ~ c + di' c- di " c^ + d^ \ c'i + d? ' * 



Note that the above operations are obtained by using the ordinary rules of algebra and replacing i^ by 
—1 wherever it occurs. 



21 



22 



COMPLEX NUMBERS 



GRAPH OF A COMPLEX NUMBER 



A complex number a + bi can be plotted as a point (a, b) on an 
xy plane called an Argand diagram or Gaussian plane. For example 
in Fig. 6-1 P represents the complex number —3 + 4i. 

A complex number can also be interpreted as a vector from 
O to P. 




POLAR FORM OF A COMPLEX NUMBER 



In Fig. 6-2 point P with coordinates (x, y) represents the complex 
number x + iy. Point P can also be represented by pola/r coordinates 
(r, e). Since x = r cos 6, y = r sin 6 we have 



6.6 



X -\- iy =^ r(cos 9 -V i sin e) 



called the polar form of the complex number. We often call r = s/x^ + y^ 
the modulus and $ the amplitude of a; + iy. 




Fig. 6-2 



lUlTIPLICATION AND DIVISION OF COMPLEX NUMBERS IN POU^R FORM 



6.7 
6.8 



[ri(cos tf 1 + 1 sin e{i\ [r2(cos $2 + * sin «2)] — »'i'"2[cos (*i + tfg) + ^ sin (ffj + e^i] 
ri(cos ffi + 1 sin ffx) '"i 

— ; : — : T — COS (81 — So) + * 31" (*1 ~ *2)l 

r2(cos e-i + I sin 62) rg ^ ^ ^ ^ ^ ■" 



DE MOIVRE'S THEOREMi|H 

If p is any real number, De Moivre's theorem states that 
6.9 [r(cos 9 A- i sin ff)]" = r''(cos pS + i sin p$) 



ROOTS OF COMPLEX NUMBERS 



6.10 



If p = 1/n where n is any positive integer, 6.9 can be written 

9 -I- 2fcff 



[r(cos ff + t sin ff)] ^■''' = r^' 



+ i sin 



* + 2k 



^] 



where k is any integer. From this the n nth roots of a complex number can be obtained by putting 
k = 0,1,2, . ..,n-l. 



7 



EXPONENTIAL and LOGARITHMIC 
FUNCTIONS 



7.2 


a^laf^ = a""' 


7.3 


(aP)i = aP" 


7.5 


a-p = 1/aP 


7.6 


{q,6)p ^ apftp 


7.8 


"/ — 


7.9 


Va/6 = yfa/yfb 



LAWS OF EXPONENTS 



In the following p, q are real numbers and m, n are positive integers. In all cases division by zero is 
excluded. 



7.1 aP' ai = aP + 'i 

7.4 qO = 1, o ?^ 

7.7 v^ - oi/« 



In aP, p is called the exponent, a is the base and op is called the pth power of a. The function y = a^ 
is called an exponential function. 



LOGARITHMS AND ANTILOGARITHMS 



If aP = N where a # or 1, then p = logo N is called the logarithm ot N to the base a. The number 
N = a'' is called the antilogarithm of p to the base a, written antilogy p. 

Example: Since 3^ = 9 we have logs 9 — 2, antilog3 2 = 9. 
The function y = log„x is called a logarithmic function. 



LAWS OF LOGARITHMS 



m^: i>m-i-l;^-mU-7''liiikV.9Ui v*p,1H^KS*« 



7.10 
7.11 
7.12 



log„ MA' ^ log„ M + log„ N 

M 

logo -^ - logo M - log„ A^ 

iogfl M^ = p logn M 



COMMON LOGARITHMS AND ANTILOGARITHMS 



Common logarithms and an ti logarithms [also called Briggsian] are those in which the base a — 10. 
The common logarithm of N is denoted by logigN or briefly logN. For tables of common logarithms and 
antilogarithms, see pages 202-205. For illustrations using these tables see pages 194-196. 



23 



24 



EXPONENTIAL AND LOGARITHMIC FUNCTIONS 



NATURAL LOGARITHMS AND ANTIIOGARITHMS 



Natural logarithms and antilogarithms falso called Napierian] are those in which the base a= e = 
2.71828 18 . . . [see page 1]. The natural logarithm of N is denoted by logg 'N or In A'. For tables of natural 
logarithms see pages 224-225. For tables of natural antilogarithms [i.e. tables giving e^ for values of ic] 
see pages 226-227. For illustrations using these tables see pages 196 and 200. 



CHANGE OF BASE OF LOGARITHMS 



The relationship between logarithms of a number 'N to different bases a and 6 is given by 

log.iV 



7.13 

In particular, 
7.14 

7.15 



logoiV = 



logb a 



loggiV = InN = 2.30258 50929 ...logioN 
logjoN = logiV =: 0.43429 44819 ...log^iV 



i 



RELATIONSHIP BETWEEN EX 



TIAL AND TRIGONOMETRIC FUNCT IONS 



7.16 e^ = cos 9 + isins, e~'^ = cos ff — i sin ff 

These are called Euler's identities. Here i is the imaginary unit [see page 21]. 

7.17 
7.18 
7.19 ' 



7.20 
7.21 
7.22 



sin e 


= 


2i 


cos tf 


= 


2 


tan 


= 




cotff 


= 






- 


2 




gie + e-" 






2i 



e« -f e-'« 



,w _ «-« 



PERIODICITY OF EXPONENTIAL FUNCTIONS 



7.23 e«»+2feTJ = fiW k = integer 

From this it is seen that e* has period 2irx. 



EXPONENTIAL AND LOGARITHMIC FUNCTIONS 



25 



POLAR FORM OF COMPLEX NUMBERS EXPRESSED AS AN EXPONENTIAL 

The polar form of a complex number x + iy can be written in terms of exponentials [see 6.6, page 22] 
*'*■* X + iy = r(cos a -\- i sin e) — re'S 



OPERATIONS WITH COMPLEX NUMBERS IN POLAR FORM 



Formulas 6.7 through 6.10 on page 22 are equivalent to the following. 



7.26 

7.27 

7.28 



r2e'"2 r2 



(rei«)p = r^e'pfl [De Moivre's theorem] 



LOGARITHM OF A COMPLEX NUMBER 



7.29 



In (re») = \nr + ie + 2kwi k = integer 




DEFINITION OF HYPERBOLrc FUNCtlONS 



8.1 

8.2 
8.3 
8.4 
8.5 
8.6 



Hyperbolic sine of a; = sinh x = 



Hyperbolic cosine of a; = cosh x = 



Hyperbolic tangent of z = tanh z — 



Hyperbolic cotangent of x = coth x — 



Hyperbolic secant of a; = sech x = 



Hyperbolic cosecant of x = csch x = 



gX 


— e~ 


X 




2 




^ + e- 


-X 




2 




e^ 


— e 


-X 


e^ + e- 


-X 


e^ 


+ e' 


-E 


e^ 


— e' 
2 


-I 


e^ 


+ e- 
2 


-X 



dI _ P--I 



RELATIONSHIPS AMONG HYPERBOLIC FUNCTIONS 



8.7 
8.8 
8.9 

8.10 

8.11 
8.12 
8.13 



sinh a; 





coshic 








1 


= 


coshsr 




tanh X 


sinhx 


sechx 


1 






cosh X 




csch a; 


1 






sinh a; 




cosh^ X 


— sinh^ X — 


1 




sech^ a; 


+ tanh2 X = 


1 




C0th2 X 


— csch^ X = 


1 





FUNCTIONS OF NEGATIVE ARGUMENTS 



8.14 sinh (—a:) — — sinh a: 
8.17 csch (— ic) = —csch a: 



8.15 cosh(— «) = cosh a: 
8.18 aech(-a;) = sech a; 



8.16 tanh (—a:) = — tanha; 
8.19 coth (-a;) - -cotha; 



26 



HYPERBOLIC FUNCTIONS 



27 




8.20 
8.21 
8.22 

8.23 



sinh {x ± y) 
cosh {x ± y) 

tanh (x ± y) 



DITION FORMUiAS 



sinh X cosh y ± cosh x sinh y 

cosh X cosh v ± sinh x sinh i/ 

tanh X ± tanh |/ 
1 ± tanh a; tanh j/ 




coth ix±y) = 



coth z coth y ± 1 
coth v ± coth X 



DOUBLE ANGLE FORMULAS 



8.24 


sinh 2x 


= 2 sinh X cosh x 


8.25 


cosh 2x 


= cosh^x + sinh^x 


8.26 


tanh 2a; 


2 tanhx 

1 + tanh'* X 



= 2 cosh2 X - 1 = 1 + 2 sinh2 x 



8.27 
8.28 
8.29 



HALF ANGLE FORMULAS 



sinh I 



cosh: 



tanh I 






cosh X — 1 



cosh X + 1 



cosh X — 1 



[+ if X > 0, - if X < 0] 



\ cosh X + 1 
sinh X cosh x — 1 



[+ if X > 0, - if X < 0] 



cosh X + 1 



sinhx 




8.30 
8.31 

8.32 

8.33 
8.34 

8.35 



sinh 3x = 

cosh 3x — 

tanh 3x = 

sinh 4x = 

cosh 4x = 

tanh 4x — 



3 sinh X + 4 sinh^ x 

4 cosh^ X — 3 cosh x 

3 tanh x + tanh^ x 
1 + 3 tanh2 x 

8 sinh^ X cosh x + 4 sinh x cosh x 

8 cosh'' X — 8 eosh2 x + 1 

4 tanh x + 4 tanh-^ x 
1 + 6 tanh2 x + tanh* x 



28 



HYPERBOLIC FUNCTIONS 




8.36 
8.37 
8.38 
8.39 
8.40 
8.41 



sinh" X 


— 


^ cosh 2ar — ^ 


cosh* X 


- 


1 cosh 2x + ^ 


sinh^ a: 


- 


i sinh 3x — 1 sinh a 


cosh^ X 


- 


^ cosh 3a; + a cosh a; 


sinh^ X 


= 


1^ — ^ cosh 2x + ^ cosh 4a; 


cosh* X 


= 


f + ^ cosh 2x + ^ cosh 4x 



8.42 
8.43 
8.44 
8.45 
8.46 
8.47 
8.48 



SUM, DIFFERENCE AND PRODUCT OF HYPEl 

sinh X + sinh i/ = 2 sinh ^{x + y) cosh ^(x — y) 

sinh X - sinh y = 2 cosh ^(x + y) sinh ^(x - y) 

coshx + coshy = 2 cosh ^(x + y) cosh ^(x — j/) 

cosh X ~ cosh y = 2 sinh ^{x + y) sinh ^(x — y) 

sinh X sinh y = ■J{cosh ix + y) - cosh {x - y)) 

cosh X cosh y — ^{cosh (x + y) + cosh (x — y)) 

sinh X cosh y — ^{sinh {x + y) + sinh (x — y)} 



XPRESSION OF HYPERBCtIC FUNCTIONS IN TERMS OF OTHERS 



In the following we assume x > 0. If x < use the appropriate sign as indicated by formulas 8.14 
to 8.19. 





sinh X = a 


cosh X = M 


tanh X — w 


coth X = u 


sech X ~ M 


csch X — M 


sinh X 
coshx 
tanh X 
coth X 
sech X 
csch X 


u 










1/m 


V«2-l 

u 


m/V1 - u^ 


1/Vk2 - 1 


Vl - "2/m 

l/7( 


Vl + W2 


1/Vl - m'^ 

M 

1/m 


m/Vm2 - 1 
1/m 

M 


Vl + uVu 


k/Vi + «2 


^/u^ - llu 


Vl-w- 


1/Vl + m2 


V"'' + 1/m 


uHv?- - 1 
1/m 


l/\/l - m2 

u 


\/l + "^ 


1/m 


Vi-«2 


Vm2 - 1/m 


m/V1 + m2 

M 


I/Vk^ - I 


Vi - uV« 


Vw2 - 1 


«/Vl - m^ 



HYPERBOLIC FUNCTIONS 



29 



)F HYPERBO 




8.49 




Fig. 8-1 



8.50 



y = cosh X 




O 



Fig. 8-2 



8.5 i 



y = tanh x 




Fig. 8-3 



8.52 y = coth X 

y 



Fig. 8-4 




8.54 y = csch X 

y\ 



Fig. 8-5 



Pig. 8-6 



JNVERSE HYPERBOLIC FUNCTIONS 



I^ X = sinhj/, then y = sinh~'x is called the inverse hyperbolic sine of x. Similarly we define the 
other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the 
case of inverse trigonometric functions [see page 17] we restrict ourselves to principal values for which 
they can be considered as single-valued. 

The following list shows the principal values [unless otherwise indicated] of the inverse hyperbolic 
functions expressed in terms of logarithmic functions which are taken as real valued. 



8.55 
8.56 

8.57 
8.58 
8.59 
8.60 



sinh->a; = In (x + y/x^ + 1 ) 



cosh""! X — In (a; + ^/x^ — 1 ) 



tanh ^ X = — In ( :; 

2 \1 — X 



coth * X = - in — -- 

2 \x — 1 



c«:h->x = In(^^+ -^^+1 



— » < « < =0 

x^l [cosh^^x > is principal value] 
-1 < X < 1 

X > 1 or X < — 1 



< X S 1 



x^ 



[sech~' X > is principal value] 



30 



HYPERBOLIC FUNCTIONS 




8.61 
8.62 
8.63 
8.64 
8.65 
8.66 
8.67 



RELATIONS BETWEEN INVERSE HYPE^G^fcFUNCTIONS 

esch-'a; = sinh~^(l/a:) 

sech-^a: = cosh^i(l/a:) 

coth-ix = tanh"' (1/a;) 

sinh~i {—x) = — sinh-i x 

tanh~*(— a;) = — tanh^'a; 

coth^i {—x) = — coth~' X 

csch^* (— ar) — — cseh~' x 




RAPHS OF INVERSE HYPERBOLIC FUNCTIONS 




8.69 y = cosh -la: 




Fig. 8-8 

8.72 y - sech-ia: 

V 



O 



II 




Fig. 8-10 



Fig. 8-11 



Fig. 8- J 2 



HYPERBOLIC FUNCTIONS 



31 



RELATIONSHIP BETWEEN HYPERBOLIC AND TRIGONOMETRIC FUNCTIONS 



8.74 sin (ix) = i sinh z 

8.77 CSC {ix) = — i csch x 

8.80 sinh [ix) — i sin x 

8,83 csch (ix) = — i CSC x 



8.75 cos [ix) — cosh x 

8.78 sec (ix) = sech x 

8.81 cosh (ia;) — cos a; 

8.84 sech (ix) — sec x 



8.76 tan (ix) — i tanh x 

8.79 cotii'x) — — icothx 

8.82 tanh {ix) = i tan x 

8.85 coth (ia:) = —i cot a: 



PERIODICITY OF HYPERBOLIC FUNCTIONS 



In the following k is any integer. 

8.86 smh(x4-2ijrt") = sinhx 8.87 cosh(x + Zkin) ^ cosh a: 8.88 tanh{x + kvi) = tanh a: 

8.89 csch (a; + 2kiri) ^ csch x 8.90 sech {x + 2kTri) = sech x 8.91 coth (a; + kiri) = coth a; 

RELATIONSHIP BETWEEN INVERSE HYPERBOLIC AND INVERSE TRIGONOMETRIC FUNCTIONS 



8.92 sin~i(ix) = i sinh"' a; 

8.94 cos ~^ X =■ ± i cosh ~ ' x 

8.96 tan-i(iar) = i tanh -'a 

8.98 cot"''(ta;) — — ieoth~'a: 

8.100 sec~'x = ±isech~^a; 

8.102 csc~' {ix) ^ ~i csch~^ x 



8,93 sinh~i {ix) = i sin~' x 

8.95 cosh"' a: = ±icos~^x 

8.97 tanh-'(ix) = itan-'a: 

8.99 coth-'(ia:) = -icot-ix 

8.101 sech~^ X = ±i sec~^ x 

8.103 cseh~'(ta;) — — icsc~ix 




QUADRATIC EQUATION: ax- -h bx + c = 



9.1 



Solutions : 



-6 ± \/b^ - iac 



2a 



9.2 



If a, b, c are real and if D = b^ — 4a« is the discriminant, then the roots are 
{i) real and unequal if D > 
(ii) real and equal if D = 
(iii) complex conjugate if Z? < 

If x^, X2 are the roots, then Xi + arg — —b/a and a;,a-2 — c/a. 



Ee 



CUBIC EQUATION; x'^ + a,x^ -\- a-iZ + a^ = 



L^ 



Q = 



Za^ ~ (i\ 



R = 



9aja2 — 27a3 — 2ai 



9 ' " 54 



a:, = S + r - lai 
9.3 Solutions: -j ajg = -^(S + T) - ^Oi + ^i-/3 (S - T) 

3^3 = -i(S + r) - !^a, - \iVZ(S - T) 

If ttj, 03,03 are real and if D = Q'^ + R'^ is the discriminant, then 
(i) one root is real and two complex conjugate if D > 
(ii) all roots are real and at least two are equal if Z> = 
(iii) all roots are real and unequal if D < 0. 

If D < 0, computation is simplified by use of trigonometry. 



9.4 



93 



Solutions if Z> < 0: 



Xi - 2\/— Q cos (J*) 

Xz = 2-/^ cos (^« + 120°) 

23 = 2^/^cos(^tf + 240°) 



where cos S = —R/yJ—Q^ 



where Xi,xi,,x^ are the three roots. 



= -a-i 



32 



SOLUTIONS OF ALGEBRAIC EQUATIONS 



33 




* B^ BM I I ■ iiii^ii l i I i llllllM^^BTCT i l ll ll il llll fcii 

QUARTIC EQUATION: x* + a,*^ + azX^ + CsX + a4 = 



Let yi be a real root of the cubic equation 
9.6 y3 ~ 02^2 -I- (a^ag - 404)1/ + (40304 -al- ofo^) - 



9.7 Solutions: The 4 roots of z^ + ^{a, ± y/a\ - ia^ + Ay^ )z + ^{j/j ± y/y\ - 404 } = 

If all roots of 9.6 are real, computation is simplified by using that particular real root which produces 
all real coefficients in the quadratic equation 9.7. 



9.8 



Kj + a^a + ajg + 3:4 = — Oi 

XiX2 + X2ir3 + X^X^ + X^Xi + X^X^ + X2X^ = a2 
XiX2X^ + ^2X3X4 + 9;j!E2-'^4 "I" ^l^Z-'^i ^^ — *^3 
3:1X2X3X4 = 04 

where Xj, Xg, Xg, X4 are the four roots. 



10 



FORMULAS from 
PLANE ANALYTIC GEOMETRY 



DISTANCE d BETWEEN TWO POINTS Piixx,yi) AND ^2(2^2,^2) 



10.1 



d = V (3^2 -Xi)2 + (j/2- 1/1)2 



:iL 



P,(a;i,!/i) 



l(3/2-!/i) 



1^2- X,) 



-1 



Xi 



Fiff. 10-1 



HOPE m OF LINE JOINING TWO POINTS Piixi,yi) AND P2{x.z,yz) 






10.2 



V2 - Vi 
m = = tariff 

X2 - «! 



UATION OF LINE JOINING TWO POINTS Pi{Xi,yi) AND Pz{x2,y2) 



10.3 
10.4 



y-y\ _ vz- vi 

X — Xi x^— Xi 



m or y — Vi = «*(« — a^i) 



y = mx + 6 



XzVi — Xiifz 
where =^ y.— mx, = is the intercept on the y axis, i.e. the y intercept. 

X2 - i'l 



lEQUATION OF LINE IN TERMS OF x INTERCEPT a^O AND y INTERCEPT b ¥= 



10.5 



- + ^ - 1 
a b 



Fig. 10-2 



34 



FORMULAS FROM PLANE ANALYTIC GEOMETRY 



35 




10.6 xcosa + ysina — p 

where p = perpendicular distance from origin O to line 

and a = angle of inclination of perpendicular with 

positive X axis. 




Fig. 10-3 




GENERAL EQ 




10.7 



Ax + By + C = 



DISTANCE FROM POINT {xi,yi) TO LINE Ax + By + C = 



10.8 



Axj + Byi + C 



where the sign is chosen so that the distance is nonnegative. 



LNGLE ^ BETWEEN TWO LINES HAVING SLOPES mi AND Mz 



10.9 



tan^ = 



Wl2 ~ Wi 



1 + mim2 

Lines are parallel or coincident if and only if wii = m2. 
Lines are perpendicular if and only if m2 =■ — iMi- 




Fig. 10-4 



slope Bii 
slope nij 



AREA OF TRIANGLE WITH VERTICES AT {a;i, ?/i), (:r2, 3/2), (a^a. ^s) 



10.10 Area = ± 



Xi Vi 1 

X2 i/2 1 

X3 Va 1 



ix2, j/a) 



where the sign is chosen so that the area is nonnegative. 
If the area is zero the points all lie on a line. 



(*i.Vi) 




Fig. 10-5 



36 



FORMULAS FROM PLANE ANALYTIC GEOMETRY 



rRANSFORMATION OF COORDINATES INVOLVING PURE TRANSLA 



10.11 



X = x' + X(i 

y - y' + vq 



JX' = X ~ Xq 

{y' = y - Vq 



where {x, y) are old coordinates [i.e. coordinates relative to 
xy system], {x',y') are new coordinates [relative to x'y' sys- 
tem] and (aro.tfo) are the coordinates of the new origin O' 
relative to the old zy coordinate system. 



O 



y' 



ixa-Vii) 



O' 



Fig. 10-6 



TRANSFORMATION OF COORDINATES INVOLVING PURE ROTATION 



x' 



10.12 



ic — a; cos a — 1/ sin a (x' — x cosa + y nma 

_ , , or -^ 

y — X sma + y'cosa yy' = ?/ cos a — x sin a 

where the origins of the old [xy] and new [x'y'] coordinate 
systems are the same but the x' axis makes an angle a with 
the positive x axis. 



.y' 



0^ 



\ 

Fig. 10-7 



TRANSFORMATION OF COORDINATES INVOLVING TRANSLATION AND ROTATION 



10.13 



X = x' cos a — y' sin a + xq 
V = x' sin a + y' cos a + y^ 

x' = (x — Xq) cos a + iy — yo) sin a 

y' = [y — Vo) cos a — (x — x^) sin a 

where the new origin O' of x'y' coordinate system has co- 
ordinates (xQ,yQ) relative to the old xy coordinate system 
and the x' axis makes an angle a with the positive x axis. 



-^ 



\ 



\ 



o \ 

\ 
\ 

Fig. 10-8 



x' 




A point P can be located by rectangular coordinates (x,^) or 
polar coordinates (r, »). The transformation between these coordinates 

Is 



10.14 



X = r cos * 
y — r sin $ 



r = V*^ + y^ 

= tan-^ (y/x) 




O 






Fig. 10-9 



FORMULAS FROM PLANE ANALYTIC GEOMETRY 



37 



EQUATION OF CIRCLE OF RADIUS R, CENTER AT [x^yo) 



10.15 



(a!-Xo)2+(wo)2 = fl2 




Fig. 10-10 



IQUATION OF CIRCLE OF RADIUS R PASSING THROUGH ORI6I 



10.16 r = 2Rcos(s-a) 

where (r, e) are polar coordinates of any point on the 
circle and (R,a) are polar coordinates of the center of 
the circle. 




Fig. 10-11 



CONICS [ELLIPSE, PARABOLA OR HYPERBOLA] 



If a point P moves so that its distance from a fixed point 
[called the focus] divided by its distance from a fixed line [called 
the directrix] is a constant e [called the eccentricity], then the 
curve described by P is called a conic [so-called because such 
curves can be obtained by intersecting a plane and a cone at 
diflFerent angles]. 

If the focus is chosen at origin O the equation of a conic 
in polar coordinates (r, e) is, if OQ = p and LM = D, [see 
Fig. 10-12] 



10.17 



P 



tD 



1 — e cos e 

The conic is 

(i) an ellipse if « < 1 
(ii) a parabola if « = 1 
(iii) a hyperbola if e > 1. 



1 — e cos 8 




Directrix 



Fig. 10-12 



38 



FORMULAS FROM PLANE ANALYTIC GEOMETRY 



ELLIPSE WITH CENTER C(Xo, 1/0) AND MAJOR AXIS PARALLEL TO x 

10.18 Length of major axis A'A — 2a 

10.19 Length of minor axis F'B = 26 

10.20 Distance from center C to focus F or F' is 



= Va2 - 62 



10.21 Eccentricity ^ 



v^airp 






O 



10.22 Equation in rectangular coordinates: 
{a: - Xo)g {y - Vp)^ 

n "r 10 — 1 



62 



10.23 




Fig. 10-13 



Equation in polar coordinates if C is at O: r^ = -r — 

a2 sin2 9 + 0* cos2 6 



10.24 Equation in polar coordinates if C is on « axis and F' is at O: r — -^ — ^ 

1 — « cos e 

10.25 If P is any point on the ellipse, PF + PF' - 2a 

If the major axis is parallel to the y axis, interchange x and j/ in the above or replace e by iir 

90° - e]. 



e [or 



PARABOLA WrTH AXIS PARALLEL TO X AXIS 



If vertex is at A{xQ,yff) and the distance from A to focus F is a > 0, the equation of the parabola is 

10.26 iy — yo)^ = ^a{z — XQ) if parabola opens to right [Fig. 10-14] 

10.27 {y~yo)^ = — 4a{ic-a;o) if parabola opens to left [Fig. 10-15] 
If focus is at the origin [Fig. 10-16] the equation in polar coordinates is 



10.28 




2a 



1 — cos 9 





Fig. 10-14 Fig. 10-15 Fig. 10-16 

In case the axis is parallel to the y axis, interchange x and y or replace 8 hy ^ir — 8 [or 90° — e]. 



FORMULAS FROM PLANE ANALYTIC GEOMETRY 




PERBOIA WITH CENTERX^o^yo) AND^AJ OR AJOSPll^rLilTOT^ 



«'n. 



^H 




Fig. 10-17 



10.29 Length of major axis A'A = 2a 



10.30 Length of minor axis B'B = 26 



10.31 Distance from center C to focus F or F' = e = yaf+b^ 



C V^2T62 

10.32 Eccentricity e = — = 

a a 



10.33 Equation in rectangular coordinates: 



(g - iCoP {y - yg)^ 
a2 62 



= 1 



10.34 Slopes of asymptotes G'H and Gff' = ±~ 



10.35 Equation in polar coordinates if C is at O: r^ = 



a262 



62 cos^ tf — a2 sin2 g 



10.36 Equation in polar coordinates if C is on X^ axis and F' is at O: r = - °^ ~ — ii- 

1 — E cos S 

10.37 If P is any point on the hyperbola, PF ~ PF' = ±2a [depending on branch] 



If the major axis is parallel to the y axis, interchange x and y in the above or replace e by i^r — e 
for 90° - el 



LSMNfSCATE 



11.1 Kquation in polar coordinates: 

7-2 — Or cos 2tf 

11.2 Equation in rectangular coordinates: 

11. 3 Angle between AB' or A'B and x axis ~ 45'' 

1 1 .4 Area of one loop — a^ 





V 




As 




/^ 


~ 




^ 


(^% 


/ 


^ 




\ 










A'y 




B' 



Fig. I 1-1 



CYCLOID 



1 1 .5 Equations in parametric form: 

X — a(0 — sin 0) 
— a{l — COS0) 

11.6 Area of one arch = Swa^ 

1 1 .7 Arc length of one arch = 8a 

This is a curve described by a point P on a circle of radius 
a rolling along x axis. 




Fig. 11-2 



HYPOCYCLOID WITH FOUR CUSPS 



11.8 Equation in rectangular coordinates: 

n .9 Equations in parametric form: 

X — a cos** 8 
y = a sin'^ 8 

11.10 Area bounded by curve = ■f"''*^ 

11.11 Arc length of entire curve — 6a 

This is a curve described by a point P on a circle of radiiis 
a/4 as it rolls on the inside of a circle of radius a. 




Fig. 11-3 



40 



SPECIAL PLANE CURVES 



41 




11.12 Equation: r — a(l + cos tf) 

1 1.13 Area bounded by curve = j^a^ 



1 1.14 Arc length of curve = 8a 

This is the curve described by a point P of & circle of radius 
a as it rolls on the outside of a fixed circle of radius a. The 
curve is also a special case of the limacon of Pascal [see 11.32]. 



lATENARY 



11.15 Equation: y = -{e^^'^+e^'^") = acosh- 

2 o, 

This is the curve in which a heavy uniform chain would 
hang if suspended vertically from fixed points A and B. 



THREE-LEAVED ROSE 



11.16 Equation: r = a cos 3« 

The equation r = a sin 3s is a similar curve obtained by 
rotating the curve of Fig. 11-6 counterclockwise through 30^^ or 
Tr/6 radians. 

In general r — a cos ns or r = a sin n8 has n leaves if 
n is odd. 




Fig. n-4 




Fig. ll-« 




11.17 Equation: r = a cos 2tf 

The equation r = a sin 2e is a similar curve obtained by 
rotating the curve of Fig. 11-7 counterclockwise through 45° or 
w/4 radians. 

In general r — a cos ntf or r ~ a sin ne has 2n leaves if 
n is even. 




Fig. 11-7 



42 



SPECIAL PLANE CURVES 



11.18 Parametric equations: 



K = (a + 6) COS 8 — b cos ( — -. — ) 



y = (a + 6) sin tf — 6 sin ( — ; — I e 



This is the curve described by a point P on a circle of 
radius 6 as it rolls on the outside of a circle of radius a. 

The cardioid [Fig. 11-4] is a special case of an epicycloid. 




Fig. 11-8 



11.19 Parametric equations: 



X = (a—b) cos + 6 cos [ — z — ) * 



y = {a—b) sin ^ — 6 sin [ — j — ] ^ 



This is the curve described by a point P on a circle of 
radius b as it rolls on the inside of a circle of radius a. 

If 6 = a/4, the curve is that of Fig. 11-3. 




Fig. 11-9 



TROCHOID 



1 1 .20 Parametric equations: 



X = atp — b sin <p 
y = a — b cos 



This is the curve described by a point P at distance 6 from the center of a circle of radius a as the 
circle rolls on the x axis. 

If b < a, the curve is as shown in Fig. 11-10 and is called a curtate cycloid. 

If b > a, the curve is as shown in Fig. ll-H and is called a prolate cycloid. 

If & = a, the curve is the cycloid of Pig. 11-2. 





Fig. 11-10 



Fig.U-U 



SPECIAL PLANE CURVES 



43 




11.21 Parametric equations: 



a; — a In (cot ^^ — cos ^) 
y — a sin 



This is the curve described by endpoint P of a taut string 
PQ of length a as the other end Q is moved along the x 
axis. 



WITCH OF AGNESI 




8aS 



11.22 Equation in rectangular coordinates: y = ■ „ -l 4„2 



1 1 .23 Parametric equations: 



X — 2a cot e 

y — a{\ — cos 2fl) 



In Fig. 11-13 the variable line OA intersects y = 2a 
and the circle of radius a with center (0, a) at A and B 
respectively. Any point P on the "witch" is located by con- 
structing lines parallel to the x and y axes through B and 
A respectively and determining the point F of intersection. 




Fig. 11-13 



FOLIUM OF DESCARTES 



11.24 Equation in rectangular coordinates; 

3[;3 4- ^3 = Zaxy 



1 1 .25 Parametric equations: 



V = 



Sat 
1 + (3 

3at^ 
1 + (3 



11.26 Area of loop 



= ^nZ 




1 1.27 Equation of asymptote: x + y + a = 



Fig. 11-14 



INVOLUTE OF A CIRCLE 



1 1 .28 Parametric equations: 

x = a(cos + sin 0) 

y = a(sin — cos 0) 

This is the curve described by the endpoint P of a string 
as it unwinds from a circle of radius a while held taut. 




Fig. 11-15 



44 



SPECIAL PLANE CURVES 



Equation in rectangular coordinates: 

(oa;)2/3 + (6y)2/3 = (a2 _ 62)2/3 



11.29 



1 1 .30 Parametric equations: 

ax = (a^ ~ 62) cogS g 

by = {a2 - 62) ginS g 

This curve is the envelope of the normals to the ellipse 
xya^ + y^/b^ = 1 shown dashed in Fig. 11-16. 




Fig. 11-16 




11.31 Polar equation: r* + a* - 2aV2 cos 29 = 6* 

This is the curve described by a point P such that the product of its distances from two fixed points 
[distance 2a apart] is a constant b^. 

The curve is as in Fig. 11-17 or Fig. 11-18 according as 6 < a or b > a respectively. 

If 6 = o, the curve is a lemniaeate [Fig. 11-1]. 



^^i 



^ 




Fig. 11-17 



Fig. 11-18 



:ON OF PASCAl 



1 1 .32 Polar equation; r = b ■¥ a cos 8 

Let OQ be a line joining origin O to any point Q on a circle of diameter a passing through O. Then 
the curve is the locus of all points P such that PQ = 6. 

The curve is as in Fig. 11-19 or Fig. 11-20 according as 6 > a or 6 < a respectively. If b = a, the 
curve is a cardioid [Fig. 11-4]. 

y 





Fig. 11-19 



Fig. n-20 



SPECIAL PLANE CURVES 



45 




CISSOID OF DIOCIES 



11.33 Equation in rectangular coordinates: 



y2 = 



2a — X 



11.34 Parametric equations: 

fx — 2a sin^ 9 



V - 



2a sin^ 9 
cos 9 



This is the curve described by a point P such that the 
distance OP = distance RS. It is used in the problem of 
duplication of a cube, i.e. finding the side of a cube which has 
twice the volume of a given cube. 




Fig. 11-21 



^SPIRAL OF ARCHIMEDES 



11.35 Polar equation: r — a9 




Fig. 11-22 



12 



FORMULAS from SOLID 
ANALYTIC GEOMETRY 



DISTANCE d BETWEEN TWO POINTS Pi{xi,yuZi) AND P%{x2,%2,Z2) 



12.1 d = \f(x^ - a;,)2 + (ya - y,)2 + (z^ - ^^)2 



Pz(^2.y2'^2) 




FifT. 12-1 



DIRECTION COSINES OF LINE JOINING POINTS P^{xx,yi,z,) AND >2(i2, 1/2,^2) 



12.2 



i — COS a = -t , m = cos/3 = , n — cos y — 

add 

where a,/?,y are the angles which line PjPa makes with the positive x,y,z axes respectively and 
d is given by 12.1 [see Fig. 12-1]. 



RELATIONSHIP BETWEEN DIRECTION COSINES 



12.3 



cos2 ct + cos^ j8 + cos2 7 = 1 or I- + nfi -\- v?- = \ 



DIRECTION NUMBERS 



Numbers L,M,N which are proportional to the direction cosines l,m,n are called direction numbers. 
The relationship between them is given by 



12.4 



I ^ 



M 



N 



y/L^ + M^ + m \/lJ+~M^Tm' ^/UVmTm 



46 



FORMULAS FROM SOLID ANALYTIC GEOMETRY 



47 



EQUATIONS OF LINE JOINING Pi{xuyuZi) AND ^2(^2, ?/2, 22) IN STANDARD TORM 



12.5 



X - xi y - j/i z - z, X- Xi y - y\ z — z^ 

or 



These are also valid if l,m,n are replaced by L,M,N respectively. 



EQUATIONS OF LINE JOINING Pi{xi,yuZi) AND ^2(^2, ?/3. 22) IN PARAMETRIC FORM 

12.6 X = X] -^ It, y — ]/] + mi, z — zj + wt 

These are also valid if t,m,n are replaced by L,M,N respectively. 

ANGLE BETWEEN TWO LINES WITH DIRECTION COSINES U.muni AND ;2,m2,«2 



12.7 



cos — ^1^2 + Wimg + »liW.2 



GENERAL EQUATION OF A PLANE 



12.8 



Ax + By + Cz + D = 



[A,B,C,D are constants] 



EQUATION OF PLANE PASSING THROUGH POINTS {Xi,yi,Zi), (3^2,3/2,22), (3:3,1/3,23) 



12.9 



12.10 



X — xi y - ifi z - zi 



= 



or 



^2 - yi ^2 - 2l 



(x-x^) + 



Zg ^ Zi Xa — Xi 
H ~ ^1 ^3 ^ a^i 



iy-yi) + 



X2 - xi vz- y-i 
X3 - Xi yg - yi 



(z - zi) = 



EQUATION OF PLANE IN INTERCEPT i^ORM 



12.11 



a 6 c 
where a,b,c are the intercepts on the x,y,z axes 
respectively. 




Fig. 12-2 



48 



FORMULAS FROM SOLID ANALYTIC GEOMETRY 



12.12 



EQUATIONS OF LINE THROUGH (xo, ?/o, So) 
AND PERPENDICULAR TO PLANE Ax + By -\- Cz + D = 



^ — a^o y - yo z~ ^0 



or X — Xq + At, y = Va + Bt, z = Zq + Ct 



Note that the direction numbers for a line perpendicular to the plane Ax + By + Cz + D = are 
A,B,C. 



DISTANCE FROM POINT {xo,yo,z,y) TO PLANE Ax + By -^ Cz -\- D = 



12.13 



Axo + Byo + Czo + D 



where the sign is chosen so that the distance is nonnegative. 



12.14 



NORMAL FORM FOR EQUATION OF PLANE 



X COS a + y cos j3 + 2 cos y = p 



where p = perpendicular distance from to plane at 
P and a,p,y are angles between OP and positive x,y,z 
axes. 




Fig. 12-3 



TRANSFORMATION OF COORDINATES INVOLVING PURE TRANSLATION 



12.15 



X = x' + Xq 

y - y' + Vo 

Z - Z' -\- Zq 



or 



= z — Za 



where (x, y, z) are old coordinates [i.e. coordinates rela- 
tive to xyz system! , {x',y',z') are new coordinates [rela- 
tive to x'y'z' system] and (kqi l/o-'^o) ^^^ ^^^ coordinates 
of the new origin O' relative to the old xyz coordinate 
system. 



A 




\z' 



Jjfo^o^^o)^ _ , 



Fig. 12-4 



FORMULAS FROM SOLID ANALYTIC GEOMETRY 



49 



12.16 



TRANSFORMATION OF COORDINATES rNVOLVING PURE ROTATION 



X = l^x' + I2V' + I3Z' 

y = Ttiix' + m^y' + m^z' 
z = n,r' + n2V' + TI32' 



x' = lix + m-iy + riiz 

y' - I2X + m^y + 7132 
s' - l^x + m^y + Mgz 

where the origins of the xyz and x'y'z' systems are the 
same and /i,7ni,?i,; i2.»«2>"2'> '3.'«3."3 are the direction 
cosines of the x',y',z' axes relative to the x,y,z axes 
respectively. 




Fig. 12-5 



TRANSFORMATION OF COORDINATES INVOLVING TRANSLATION AND ROTATION 



12.17 



= hx' + l^y' + hz' + aro 
— «,^' -J- ™_„' -f ^^2' 

+ ngz' + Zo 



a; = l^x' + /gy' + l^z' + 
1/ = Wia;' + m2y' + wigz 



+ I/O 



y' = ya: - Xq) + m2{y - yg) + n^iz - «o) 
^ 2' = ^3(3; - »(,) + ?n3(y - yo) + 713(2 - Zq) 

where the origin O' of the x'y'z' system has coordinates 
(a^o. Vd' ^o) relative to the xyz system and ly, m^, jj,; 
^2>*"2."2! h''^S'i^z ^^^ the direction cosines of the 
x',y',z' axes relative to the x,y,z axes respectively. 




/ 
/ 



T 



CYIINDRICAL COORDINATES {r,e,z) 



Fig. 12-6 



A point P can be located by cylindrical coordinates (r, 9, z) 
[see Fig. 12-7] as well as rectangular coordinates (ir.y, 2). 

The transformation between these coordinates is 



12.18 



X = r cos 

y = r sin ( 

z = z 



r — y/x^ + j/2 
e — tan- 1 {yfx) 
z = z 



(^.tf.z) 




Fig. 12-7 



m 



FORMULAS FROM SOLID ANALYTIC GEOMETRY 



SPHERICAL COORDINATES {r,e,,p) 



A point P can be located by spherical coordinates (r,e, 
[see Fig. 12-8] as well as rectangular coordinates (x, y, z). 

The transformation between those coordinates is 



12.19 



<ar 



X = r sm e cos <f> 
y = r sin e sin 
z — r cos e 



'r = V^2T^2 + ^ 
■^ = tan^i (j//ar) 




EQUATION OF SPHERE IN RECTANGULAR COORDINATES 



1 2.20 (X - xo)2 +{v- vo)^ + (z- zo)2 = R^ 

where the sphere has center {xq, i/q, zq) and radius R. 




Fig. 12-9 



EQUATION OF SPHERE IN CYLINDRICAL COORDINATES 

12.21 7-2 _ 2rorcos{o-9Q) + r^ + (z-Zq)^ = R^ 

where the sphere has center (r(,,tfo,«o) in cylindrical coordinates and radius R. 
If the center is at the origin the equation is 

12.22 r^ + z^ = iJ2 

EQUATION OF SPHERE IN SPHERICAL COORDINATES 



12.23 r^ + To — Zr^r sin 9 sin So cos (^ - <Pq) = R~ 

where the sphere has center {7*o, 6q, <Pq) in spherical coordinates and radius R. 
If the center is at the origin the equation is 

12.24 r = R 



FORMULAS FROM SOLID ANALYTIC GEOMETRY 



51 



EQUATION OF ELLIPSOID WITH CENTER (xo,yo,2o) AND SEMI-AXES a,b,C 



a^ b^ c^ 




Fig. 12-10 



ELLIPTIC CYLINDER WITH AXIS AS z AXIS 



12.26 



where a, 6 are semi-axes of elliptic cross section. 

If h — a it becomes a circular cylinder of radius a. 




12.27 



Fig. 12-n 



riC CONE WITH AXIS AS z AXIS 



a2 ^ 62 




HYPERBOLOID OF ONE SHEET 



Fig. 12-12 



12.28 



^ A. y^ - ^ - 1 

a2 "^ 62 c2 




Fig. 12-13 



52 



FORMULAS FKOM SOLID ANALYTIC GEOMETRY 




PERBOLOID OF TWO SHEET 



12.29 



a2 62 c2 ~ 



Note orientation of axes in Fig, 12-14. 




Fig. 12-14 




ELLIPTIC PARABOLOI 



12.30 



x2 ^ 
a2 "^ 62 




Fig. 12-15 




12.31 4-tf = - 

Note orientation of axes in Pig. 12-16. 




Fig. 12-16 



1 




DEFINITION OF A DERIVATIVE 



If 1/ - f{x), the derivative of y or f(x) with respect to 2 is defined as 

13 1 ^ = lim /<" + fc)-^<'> = lim /(^ + ^^)-/(^) 

da; ji-o ft Ax-*o Az 

where h = ^x. The derivative is also denoted by y' , df/dx or f'{x). The process of taking a derivative is 
called differentiation. 




GENERAL RULES OF DIFFERENTIATION 




In the following, u, v, w are functions of x; a, b, c, n are constants [restricted if indicated]; e = 2.71828 . . . 
is the natural base of logarithms; In m is the natural logarithm of u [i.e. the logarithm to the base e] where 
it is assumed that a > and all angles are in radians. 

13.2 £(c) = 

13.3 4-i<^=^) = « 
ax 

13.4 ^-ica;") = mcx"-> 
dx 



13.5 
13.6 
13.7 
13.8 
13.9 



d , ^ ^ ^ V du . dv dw 

da; ' dx dx dx 

d ^ . du 

(cw) = e-y- 



dx 



' dx 



d , , dv ^ du 



dx' 



dx 



d , , dw , dv . du 

-;—{uVw) ~ UV-T' + UW-i r VVJ-j— 

dx dx dx dx 



d_fu 

dx \v 



v(du/dx) — u(dv/dx) 



13.11 ^ = ^^ (Chain rule) 
dx du dx 



13.12 ^ =^ jV 
dx dx/du 



1313 — = ^^^^^ 
dx dx/du 



53 



DERIVATIVES 



DERIVATIVES OF TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNC 



fi lA d . du 

Ii».l4 ^— sinjt = cosM^— 

dx ax 

13.15 -T—cosu = — sinw-;— 
dx dx 

13.16 -J— tanw — sec^M^— 
dx dx 



13.17 3— COtU = — CSC^M-r- 

dx dx 

13.18 j--secM — secMtanu-j— 

13.19 3— cscM = — cse M cot It -r- 

dx dx 



13.20 ^sin-i« = 



1 du 



13.21 -I— cos ' u = 

dx 



:-l,i — 



13.22 Atan-.t ^ T^f^ 
(13: 1 + 1*2 f^x 



~2< sin '«< 2 



-T- [0 < cos J u < ■n\ 



-- < tan-i w < - 



13.23 £eot-„ = ^1 10<cot-.«<.] 



13.24 ^sec-.« = 



du 



±1 dw 



13.25 



dx 



csc~^u = 



\u\ Vm^-I t'* mVm2^ '^^ 

—1 dw q:l du 



\u\ y/u^— 1 



da; 



tVw^ - 1 



da: 



+ if < sec-iM < jr/2 

- if n-/2 < sec-^M < TT 

- if < csc^u < tt/Z 
+ if -7r/2 < csc-^u < 



DERIVATIVES 01^ 



PanW 



OGARIfHMIC FUNCTIONS 



,**, d , ^ogg^ dtt 

13.26 -3— log„M = -3- 

dx u dx 



1 3.27 -r- In M = -j- log. m = , 

dx dx ^^ u dx 



1 du 



13.28 :^a" = aMna^ 
da: rfa; 



13.29 :^c" - e"^ 
da; da; 



1 3.30 ^u^ =. A ew In « = e^^^^^[v\n u] 



dx 



dx 



,du, , dv 
dx dx 



»IRIVATIVES of HYPERBOLIC AND INVERSE HYPERBOLIC FUNCTIOI 



13.31 -^— sinh u 
dx 



, dv, 

cosh u -J— 

da; 



1 3.34 -=- coth u — 
dx 



, ,, du 

— csch^ M 3- 

dx 



13.32 :^coshM 
da; 



. , du 

sinh u 3- 

aa; 



13.35 3- sechw = 
dx 



~ sech M tanh u 3— 
da; 



13.33 3— tanh « 
da: 



, , du 
da; 



13.36 -— cschM — 
da; 



— csch u coth tt 3— 
dx 



DERIVATIVES 



55 



13.37 Asinh-iM 
ax 



13.38 3— cosh~^M 
dx 



13.39 -^t&nh-^u 
ax 



13.40 ~coth-iu 
dx 



13.41 ^sech-iw 
ax 



13.42 -^csch-iu 
da; 



1 du 



±1 dw 
1 f^" 

1 - m2 dK 

1 du 
l--u:idx 

^1 du 
uVT^^^^ 
-1 du 



r+ if cosh-^ M > 0, u > 1"| 
[- if cosh-iM< 0, M> ij 

[-l<u< 1] 



+1 dit 



Imj y/TT^ ^ MvTT^ '^ 



[m > 1 or u < -1] 

r- if sech-i w > 0, < u < 1"| 
L+ if sech-i M < 0, < w < ij 

[- if M > 0, + if u > 0] 



HIGHER DERIVATIVES 



The second, third and higher derivatives are defined as follows. 

13.43 Second derivative = ±(^^ = ^^ = fix) = y" 

13.44 Third derivative - ^(0) = ^ = /'"(*> = V 

jr /jn— 1., 

13.45 nth derivative 



- dx \dx^-i dx« ^ ^''* ^ 



= -. ^fl. ^'Va^.t- 



k. irfiB iV •^^ -t-fc TTff T 'tf^^^^t'\— iT^T^^l^ WT ^^ .^^^ 



LEIBNITZ'S RULE FOR HIGHER DERIVATIVES OF PRODUCTS 

Let Z?" stand for the operator 3—5- so that D'm = -^-5 = the pth derivative of u. Then 

ax'^ ax 




13.46 



■where ( 1, ( J, -.- are the binomial coefficients [page 3]. 



As special cases we have 
13.47 



^ I \ d^ , „ dudv , d^u 



13.48 



dx^ 



d^v 



du d^v . „ d^u dv . d?u 



Jz—( \ — _u 9 . ^ 4- Q . A- 



d«3 



dat^ rfa; da;^ dx^ dx di^ 



DIFFERENTIALS 



Let y = f{x) and ^y = f{x + &.x) - f{x). Then 
13.49 



Ax iz ^ dx 

where e -* as ix -• 0. Thus 

13.50 Ay 33 /'(x)Ax + *Ax 

If we call Ax = dx the differential of x, then we define the differential of y to be 

13.51 dy = f'{x)dx 



56 DERIVATIVES 

RULES FOR DIFFERENTIALS 

The rules for differentials are exactly analogous to those for derivatives. As examples we observe that 
'3-52 d{u±v±iv± ■••) = du :t dv ± dw ± ■■■ 

'3-53 d{uv) ~ udv + vdu 

1 3.54 d f?i\ - ^^dw — udv 

13.55 d{M") = nw-idu 

' 3-56 d(sin u) = cos « du 

'3.57 rf(cosM) = — sin udu 



PARTIAL DERIVATIVES 

Let fix, y) be a function of the two variables x and y. Then we define the partial derivative of f(x, y) 
with respect to x, keeping y constant, to be 

13.58 ^ ^ lim /(» + Ax, y) - fix, y) 

dZ Al-*0 Ax 

Similarly the partial derivative of fix,y) with respect to y, keeping x constant, is defined to be 

13.59 il = li^ /(x, y + Aj/) - fix,v) 

^y Ay-.0 Ay 

Partial derivatives of higher order can be defined as follows. 

13.60 ^ = A. f^] ^ ^ ±(^ 

dx^ dx \BxJ ' dy2 sy [dy 

13.61 -^ = A/a/"\ j^ = ±f^ 

dxdy dx ydyj ' SyBx 3y \Sx 

The results in 13.61 will be equal if the function and its partial derivatives are continuous, i.e. in such 
case the order of differentiation makes no difference. 

The differential of fix, y) is defined as 

13.62 df = fdx + ^dy 

ax dy 

where dx = Aa; and dy = Ay. 

Extension to functions of more than two variables are exactly analogous. 




DEFINITION OF AN INDEFINITE INTEGRA1 



li ~-= fix), then y is the function whose derivative is f(x) and is called the anti-derivative of /(ar) 
dx " -^ /• dy 

or the indefinite integral of f{x), denoted by | f{x)dx. Similarly if y - \ f{u) du, then ^ = f{u). 

Since the derivative of a constant is zero, all indefinite integrals differ by an arbitrary constant. 

For the definition of a definite integral, see page 94. The process of finding an integral is called 
integration. 




In the following, u, v, w are functions of x; a, b, p, q, n any constants, restricted if indicated; 
e = 2.71828. . . is the natural base of logarithms; In u denotes the natural logarithm of u where it is assumed 
that M > [in general, to extend formulas to cases where w < as well, replace In u by In |m|]; all angles 
are in radians; all constants of integration are omitted but implied. 



14.1 
14.2 
14.3 
14.4 

14.5 
14.6 

14.7 
14.8 

14.9 



■ dx = ax 



I af{x) dx = a \ f{x) dx 

I (u±v±w:ii ■ ■ •)di£ = I udx ± I vdx ± \ w dx ± 

i udv — uv — I vdu [Integration by parts] 

For generalized integration by parts, see 14.48. 

j f{ax) dx = — \ /(«) du 

/w" + 1 
Wdu = -^V-T» K^-1 [Forn = -l, see 14.8] 



r — - In M if «> or In {-u) if u < 
J u 

= In ImI 



du = e" 



14.10 i a"-du = I e"'" 



1 du 



gU In a 
In a 



In a ' 



a> 0, a¥^l 



57 



INDEFINITE INTEGRALS 



14.11 


1 sin u du 


= 


— cos u 


14.12 


\ cos u du 


= 


sin u 


14.13 


1 tan u du 


= 


In sec u = — In cos u 


14.14 


1 cot u du 


= 


In sinu 


14.15 


\ sec u du 


= 


In (sec tt + tan «) — In tan ( o" + T 


14.16 


1 CSC u du 


= 


in (cBC u — cot u) — In tan — 

2 


14.17 


1 sec^ u du 


- 


tanw 


14.18 


1 CSC- K du 


^ 


— cot u 


14.19 


1 tan^ u du 


= 


tan u — u 


14.20 


1 cot^ It du 


= 


— cot M — M 



_ . _, r . « . w sin 2« , , . 

14.21 I sin^K du = o" 2 — — ^(w — sinu cosu) 

1 4.22 j cos2 II du = — -\ 2 — — ^(m + sin u cos u) 

14.23 Jsec»tan«du = secu 

1 4.24 I CSC II cot u du = — CSC u 

1 4.25 I sinh tt du = cosh u 

14.26 I coah udu = sinhu 

14.27 I tanhitdw — In cosh m 

14.28 I coth It du = In sinh u 

14.29 I sechttrfw. — sin-Mtanhw) or 2 tan"* e« 

14.31 I sech- w d« — tanh u 

14.32 j csch2 urfu = -cothii 

x 



14.30 i csch u du = In tanh ^ or — coth-'c" 



14.33 I tanh^w du - u — tanhw 



INDEFINITE INTEGRALS 



59 



14.34 
14.35 
14.36 
14.37 
14.38 
14.39 
14.40 
14.41 
14.42 
14.43 
14.44 
14.45 

14.46 
14.47 
14.48 



I coth2 u du = u — coth u 

I sinh^ u du = — r -^ — ^(sinh « cosh u — u) 

1 cosh^ udu — \- — — ■|(sinh u cosh M + «) 

I sech M tanh « du = — sech u 

I csch u coth u du = — csch u 

f du _ 1 _,M 

J u2 + a2 ~ a ^" a 

/du 1 - /u — a\ 1 ^. , " 9^5 

-5 ^ = TT 1" ( . ~ ~-coth-i- m2 > o2 

u^ — a?- 2a \u + aJ a a 

/ 
/ 



^ oin 1 — 



du 

. '^" = In (m + Vw2 + a2 ) or sinh-i - 

Vu2 + a2 " 



r ^" ^ In {« + Vm2 - a2 ) 

— , — -sec~M- 

wV^^23^ a I a 

r dw ^ _l^^f a + ^u^ + a^ 
J uyju^ -t- a2 » " V tt ^ 

r du ^ _\^^{°:1S^^Z^ 



This is called generalized integration by parts. 



^'''dx 



IMPORTANT TRANSFORMATIONS 



Often in practice an integral can be simplilied by using an appropriate transformation or substitution 
and formula 14.6, page 57. The following list gives some transformations and their eflfects. 



1 4.49 r F{ax +b)dx = - f F(«) du 

1 4.50 r FcVax + b )dx = | J « ^(w) du 

14.51 r Fi^fax + 6 ) dx = - f w'«-iF(M)du 

14.52 f F(Va2 -x'^)dx = o J F(a cos w) cos « 

14.53 j F(y/W+~^)dx ^ « ) F(a sec u) see2 « 



du 
du 



where u = ax+h 

where u = •\Jax+ 6 

where u = v**3J + ^ 

where x = a sin u 

where x — a tan u 



1* 



60 



INDEFINITE INTEGRALS 



14.54 J F{\Jx^ — a?- ) dx = a \ F{a tan u) sec u tan u du where x = a sec m 



14.55 



fFie^^)d. = If 



F(u) 



du 



1 4.56 J F{ln x)dx - ( F(u) e« du 

14.57 fFUin-i^j (te = orF{w)cosw 



du 



where u = e"^ 



where u = \nx 



where u = sin^^ — 
a 



Similar results apply for other inverse trigonometric functions. 
14.58 f Fisinx,co.x)dx - 2Jf(^^,{^\^ where u 



= tang 



Pages 60 through 93 provide a. table of integrals classified under special types. The remarks given on 
page 57 apply here as well. It is assumed in all cases that division by zero is excluded. 



14.59 
14.60 
14.61 
14.62 
14.63 
14.64 
14.65 
14.66 
14.67 
14.68 
14.69 
14.70 
14.71 



INTEGRALS INVOLVING &x + b 

7~r = - In (ax + b) 

ax + a ^ ' 

Jxdx _ X b 
r-r =: 5 In (ax + 6) 
ax + b a a^ ^ ' 

f x^dx _ {ax + 6)2 2b(ax + 6) , 6^ 

I r-r ~ ~ — „ ■ ' i — 5 H ^ In (ace + 6) 

J ax + b 2a3 a^ a^ ' 



/ 
/ 

/ 



^ (ax + &)3 _ Sb(ax + &)2 ^ BbHax + b) _ ^^^ ,^^ ■ ^ 



3a* 



x^ dx 
ax + b 

dx _ i^ I w 

x{ax + b) b \ax + b 



dx _ 1 a^ / ax+b 

zHax + b) ~ ~bx "^P^^l X 



dx 



x^{ax + 6) 

dx 
(ax + 6)2 

dx 



2ax-b o2 
262^2 ^ 63 '" V ax + 6 



-1 



J (ax 

/x' 
(ax 



a(ax + 6) 
6 



+ 6)2 
2 da; 



9, _,_ , , + — 5 In (ax + 6) 
a^ax + 0) o2 



ax + b 

i3 



62 



+ 6)2 a3 a3(aa; + 6) a^ 

x^dx _ (ax + 6)2 Bb(ax + b) 



26 , , , ,, 
5 In (ax + 6) 



(ax + 6)2 
dx 



J a;(o3; + 6)2 



2a^ 

] 

6 (ox 



+ 



63 



a*{ax + 6) a 



+ ^ In {ax + 6) 



! + 6) "^ 62^"(^aa: + 6J 



a;2(ax + 6)2 62(aa:+6) 62a: 6^ 



1 , 2a , / ttic + 6 
+ TTT In 



INDEFINITE INTEGRALS 



61 



14.72 
14.73 
14.74 

14.75 

14.76 

14.77 

14.78 

14.79 
14.80 
14.81 

14.82 



dx ^ {ax + 6)2 3a{ax + b) 

xHax + 6)2 26*x2 "^ Mx 



_ 3a^ , / ax + b 



bHax + b) b* \ 



dx 



-1 



2((m; + 6)2 
-1 



aHax + 6) ^ 2aHax + 6)2 
o3{ffla: + 6) 2aHax + 6)2 ^ "^3 '" ^'^^ + "' 



362 



63 



a? a'^{ax + 6) 2a4(ox + 6)2 a 



T In iax + 6) 



a^x< 



2aa 1 , / aa; + 6 

263(a» + 6)2 63(a« + 6) 63 I x 



S 
X 
/ 

/ 

/ 

J 

J 

J 
J 

If n ^ -1, -2, see 14.62, 14.67. 

J 

If 71 = -1, -2, -3, see 14.61, 14.68, 14.76. 

r r-r- + — \ r^ I a!'"(aa; + 6)" i dx 

m + m+1 wi + n + lj ' 



(ax + 6)3 

xdx 

{ax + 6)3 

a;2(ja: 
{<u; + 6)3 

(ax + 6)3 

dx 

x{ax + 6)3 

dx 

xHax + 6)3 

rfx 



x3(ax + 6)3 

(ax + 6)" dx = 



_ — <t 2a 1 , 3a . f ax + 6 

262(ax + 6)2 63(ax + 6) 63x 6* ^" \^ x 

a-'x2 4a3x _ {ax + 6)g _ ^ ,„ / ax + b \ 

ax + 6)2 65(ax + 6) 265x2 b^ \ x ) 



265( 

(ax + 6)" + i 
(n+l)a ■ 



If n = -1, see 14.59. 



x2(ax + 6)"dx - io-^ + bY*' _ 26(ax + 6)n-H2 62(ax+6)"-^' 
(n+3)a3 (Ti + 2)a3 ^ (n + l)a3 



14.83 



J x'"(ax + 6)"dx 



x"'(ax4-6n + * -mb T „ ,, , t. ■, 

-^-^ , ,. - 7 — ; r-rr I «'"~Vax + 6)"dx 

(m + n + l)a (m + m + l)a J ' 

:n+l)6 ^ (n + l)6 J "" (ax+ft) rfx 



M.84 J 

/ 



dx 



14.86 



yax + b 

dx 
■\/ax + 6 

x2 dx 
Vax+6 



INTEGRALS INVOLVING y/axTh 
Zyjax + 6 



2(ax-26) / — TT 
3a2 ^°^+^ 

2(3a2a:2 _ 4a6x + 862) _ 

-^ -—- -' Vox + 6 

15a3 



14.87 



14.88 



J 
J 



dx 
xyax + 6 



1 /v'ax+T — ■/& 

■/& \v'ax+6 + /6, 
2 



tan 



dx 

x^ax + 6 



[ax - 
6a! 26 J 



^/=:6 

Vax + 6 



6 



dx 



x-\/ax + 6 



[See 14.87] 



62 



INDEFINITE INTEGRALS 



14.89 
14.90 
14.91 

14.92 

14.93 
14.94 

14.95 
14.96 
14.97 

14.98 

14.99 

14.100 

14.101 

14.102 

14.103 

14.104 



ax + b dx = 



I xVax + b dx — 

I x^y/ax + b dx = 
/" y/ax + b 

r \/ax- 
J ^ 

J' g"* 
VcuF + 6 

/ 



2V(ax + 6)3 
3a 



X 

■\fax-¥h 



dx = 
dx = 
dx = 



2(15a2a;2 - 12abx + 862) . 
2yiax + 6 + 6 f 



dx 



xvax+6 
Vax + 6 a /• d« 



2 J a:. 



"^ -' ■^•\/ax+ 6 



[See 14.87] 
[See 14.87] 



2x'«V«a: + 6 2m6 T a;*"- ^ 



a J 



dz 



rfa; 



x^yax + 6 



ax + 6 dx = 



dx = - 



dx = 



\ x"'V 

/■\/ax-¥h 

/y/ax + b 
x^ 

r (ax + 6)'"/2 dx 
f x(ax + 6)'«/2(ia: 
j xHax + 6)'n/2 dx 

r^ax+_6r^ 

f (gg + b)"'^2 
J 

r dx 

J x(ax + 6)"'/2 



{2m + l)a {2m + l)a J ^^ + (, 

Vaas + 6 _ (2wi - 3)a /" dx 

*(m-l)6x'"-i {2m~2)6j a^^-iVo^Tfr 

= ,o ^Tox (t^ + &P^' - /o ^^',^ r a;"*" Vttg + 6 dx 
(27n + 3)a^ ' (2m+3)aJ 

V ax + 6 



s r 



dx 



(m-l)x'"-i 2(m-l) J a,m-iV^^T6 
-(ax + 6)3/2 (2m - BW r Vox + 6 



(m-Dbx""-! (2m 

^ 2(ax + &)t'" + 2>/2 
a(m + 2) 



-5)a f 
-2)6 J 



X"!-! 



dx 



_ 2(ax+6)tw» + 4V2 ^ 26(ax+6)t'" + 2)/2 

a2(m + 4) a2(m + 2) 

2(ax + 6)t'"+g>^2 46(ax + 6)<'"+<)/2 262(aa; + 6)<"^+2J/2 

a3(m + 6) aH7n + 4) a3{m+2) 



dx = 



dx = 



2(ox + 6)'"/2 



J 



(aX + 6)<'n-2J/2 



+ 6 I i;:::^-^^^^- — dx 

m J X 



_ (aa; + 6)t"'+2)/2 mo r (ax + 6)"'/a ^^ 



6x 
2 



(m - 2)6(ax + 6)('«-2>/2 



26 

+ 



IS 



dx 



x(ox + 6)<"'-2>/2 



INTEGRAIS INVOLVING ax -\- h AND px + g 



14.105 J 

14.106 ^ 

14.107 f 

14.108 r 



dx 



14.109 



J 



{ax 


+ h){vx + g) 




xdx 


(ax 


+ 6)(px + g) 




dx 


(ax 


-f- 6)2(px + g) 




xdx 


(ax 


+ 6)2(px + g) 




X2dx 



In 



px + q 



bp — aq \ ax+ 6 



-r-^ i~ln(ax + 6) - ^ In (px + g) 

6p — a^ a p 



+ 



In 



6p — aq 1 ax + 6 bp — aq V ox + 6 



px + q 



1 



In 



ax + 6 



bp — aq 1 6p — aqr \px + q 

6» 1 



o(ax + 6) 



+ 



(ox + 6)2(px + q) (bp - aq)aHax + 6) {bp - aq)^ \ p 



^\n{px + q) + 



^fc^hn(ax + 6) 



INDEFINITE INTEGRALS 



14.110 



/ 



dx 



-I 



dx 



(ax + h)^{px + q)" (n - l)(6p - aq) ]{ax + 6)'"^Hp3: + 9)"~^ 

+ a(m + n-2) J^^^^jj^^p^^^j^^, 

14.111 1^ 7—dx = — + ^ „ In (px + q) 

J px + q p p2 



' rl J(°^+^)"'^' + in-m-2)a C -^^^^^dx] 

{n-l){bp-aq)\{px + q)"-^ ' J (px + «)" ^ J 



14.112 



.) ipx + q)^ 



dx — 



Zl__ji^±^ 4- mibp-aq) f 

(n — 7n — l)p UpK + g)"^! ^ 



(px + 7)" 



dx 



(n-Dpltpx + g)"-! '"''J (px + g)"->'*^ 



INTEGRALS INVOLVING v/aa; + b AND px + q 



14.113 f P^ + ^ dx = 2(apa' + 3ag-2M^/^^:^ 

-* ^/^.x + h 3a2 



14.114 



/ 



dx 



'yjp(ax + b) — y/bp — 



aq 



In 



(px + g) y/ax+ b 



y/bp — ag Vp V vp("^ + 6) + \bp — 



aq 



\aq — bp vp 



tan 



-1 I p(oa; + 6) 
"^ a? — 6p 



14.115 



/ 



\ax + b 
px + q 



dx = 



2\ax + b vbp — aq lyJp{ax + 6) — yjbp — aq 
+ =— In 



P pVp \yjp{ax + 6) + y/bp — aq^ 

2yax + b 2y/aq — bp 



pVp 



tan 



_, / p(qg - 
"Y ag - 



+_61 
6p 



2(px -r qy ■ 'y/ax-ro hp - aq f 

ij>:c + qYy/ax + bdx = ^^^^fs)^ + (2« + 3)p J ^/^^^ip^ 



14.117 



r dx_ 

-^ (px + q)"- V 



2(px + g)« + JVax + 6 ^ 6p - ag r ( pg + ?)" 

/ax + 6 

(2n - 3)a T 

-6p) J 



dx = ,„ ■ „ -- h T^ - , ».'' ■ I 'T ' "" ■ da; 

(px + g)-V^^Tt ~ Tn-l){aq-bp)(px + qY~^ ' 2{n - l)(aq - bp) J (pa; + g)"-> V^^+6 



(2n + 3)p 
Vax + b 



^;i^ + 



dx 



lAllfi r <Pa^ + 9)" ^ _ 2(px + g)"Vgx+6 2n(ag-6p) r (px + g)"-idx 
■ J ,/:r7XT ''"^ - {2n + l)a + (2^+1)^ J ,/— Tft 



Vox+fe 

Va^ + 6 
(px + g)" 



—\ax + b 



V ax + 6 
dx 



,,o r V "3^ -^ o _ -V«'^'^P g r dx 

**■"' J(px + a)" "^ ^ («-l)p(p:r + g)'«-i "^ 2(7i-l)pJ (p^ + ^j„-i^/^^:jrb 



"~^^ ^ - ^^ UMI I il 

INTEGRALS INVOLVING y/ax + b AND v^xTg 



14.120 



/ 



dx 



■\/(ax + 6)(px + q) 



- In (y/a(px 4- g) + y/p{ax + b) ) 

2 



tan-i 



'—ap 



V —p(ax + 6) 
a{px + g) 



4^f 



14.121 r 



xdx _ V(aa: + b)(px -f g) fey + 



ag 



\/(ax + 6)(px + g) **P ^"P '^ ^/ (ax + b)(px + q) 



J 



dx 



U- 



64 



INDEFINITE INTEGRALS 



14.122 
14.123 
14.124 



|V{a-+6){px+,)d. ^ '"p^ \l^ + "^ V(a. + &)(px + g) - ^''^g-;;^^' 1^ 



da; 



V(aa: + b){px + g) 






VX+ q . _ Vi'^ + &)fP=g + g) ^ gg-ftp 



ax + b 



dx 



dx 



^ 2a 

2^/axTb 



f 



dx 



y/{ax + h){px + q) 



{px + q) V(aa; + 6)(pa: + g) (ag — 6p) Vpa; + q 



14.125 


J a:2 + tt2 


14.126 


C xdx 

J ar2 + a2 " 


14.127 


C k2 dx 
J x^ + a^ " 


14.128 


r x^dx 
J x^ + a^ ~ 


14.129 


r dx 

J x{x^ + a^) 


14.130 


C dx 

J ic2(a;2 + a2) 


14.131 


f dx 


14.132 


f dx 

J {ic2 + o2)2 - 


14.133 


(* xdx 

J {x2 + a2)2 " 


14.134 


r x^dx 

J (x2 + a2)2 - 


14.135 


f x^dx 
J (a;2 + a2)2 " 


14.136 


r dx 

J ar(a2 + o2)a 


14.137 


r dx 

J a;2(a;2 + o2)2 


14.138 


r dx 
J ic3(x2 + a2)2 


14.139 


f dx 

J (a:2 + a2)n 


14.140 


/• xdr 


J (a:2 + a2)n 


14.141 


z' dx 


J X(x2+a2)n 


14.142 


r ar"' rfa; 
J (a;2 + 0.2)71 


14.143 


r da; 



CEGRALS INVOLVING a;^ + 




- tan-i — 
a a 



= iln{a;2 + a2) 



X — a tan~^ — 
a 

y2 /,2 

|--|ln(x2 + a2) 



2a2^" 1x2+ a3 



■ -5 i tan 1 — 



2a2x2 2a4 ^"^ l^x^ + a^ 
a; + J^ tan-i- 



2(x2 + a2) 

— z , _1_ *an-i — 

2(x2 + a2) 2a a 

5^ + - In (x2 + a2) 

2(x2 + a2) 2 ^ ^ 



1 + A.i„ ^ ^^ 

2aHx^ + o2) "^ 2ai \x^ + a^ 

-^- ^ - -3-tan-i- 

a'fx 2a4{x2 + a^) 2a5 a 

1_ _ 1 _ J, , / xM 

Za*x^ 2aHx^ + a^) o* \^x2 + a2y 






rfx 
(x2 + a2)''-i 



2(m-l)a2(ar2 + a2)"-i (2n 

^1 

2(ri-l)(x2 + a2)n-i 

^ 1 + J^ (' dg 

2(Ti-l)a2(x2+a2)n-i ^ a2j x(x2 + a2)"-i 

r a;'"-2dx _ 2 r x'"-2dg 
J (x2 + a2)n-i " J (x2 + a2)" 

J_ r dx \_ C dx 

0.2 J a.m(3;2 + a,2)n-i ^2 J a:">-2(a;2 + tt2)n 



INDEFINITE INTEGRALS 



65 




14.144 
14.145 
14.146 
14.147 
14.148 
14.149 
14.150 
14.151 
14.152 
14.153 
14.154 
14.155 
14.156 
14.157 
14.158 
14.159 
14.160 
14.161 
14.162 



r dx 

J 3:2 - a2 - 



■dx 
- a? 



1 , f x - a \ 1 . , , a: 

2a \x + a J a a 



pn(x^-a^) 



I o, x — a 
a: + ^ In , 

2 Va: + a 



C x^dx 
J a;2 _ „2 - 

r x^dx 
J a;2 — a2 

/ dx 
iB(a;2 — a2) 

C dx _ _1_ , J_ , / x-a \ 

J xHx^ - a2) a^x "^ 2a3 \x + a/ 

^ x^{x^ — a2) 

J (g2-a2)2 

f gdg _ 

J a;2rfa; _ 
(g2 - a2)2 

(x2-a2)2 



y + yln{g2-«2) 



1 , /g2_(i2 



^ -J. In 



2a2a:2 2a^ 1x2 - a2 



- A: In / '^ - '^ 



-1 

2{g2-a2) 



+ -lln/=«-'^ 



2(a:2-o2) 4a \x + a 



~^^ -i- 1 In {g2 - o2) 



rfg 



2{x2 - (i2) 2 

-1 , 1 



+ ^In 



2a2{a;2 - qS) 2a4 Xx"^ - ti^ 



- T%ln 



a^a; 2a4{x2 - a2) 4a5 \. g + a 



+ 4ln 



2a^g2 2a4(a;2 - a2) a« Kx-^ - a 



-3B 



2n- 



J g{g2-a2)2 

C dx 

J g2(;c2-aa)2 

r dx 

J a;3(a;2_a2)2 

r dg 

J (g2-a2)n 

/ g dg _ 

{g2 - a2)« 

r dg , 

J g(g2 — a2)" 

/ g"* dg _ 

(g2 - a2)n 

r dg ^ 1 r dg \^ r 

J g'"(g2-a2)n a2j a;'"-2(g2 - a2)n a2 J 



2(Ti-l)a2(g2-a2)"-i (2n 

-1 



t-3 r dx 

-2)o2j (a;2-a2)"-l 



2(n-l)(g2-a2)"-i 

-1 J, r dg 

-I)a2(g2_a2)n-1 a2j x(g2-a2)"-i 



2(n 



J g'"-2dg 2 r g"'-2dg 

(g2-a2)n-I ^ " J (g2-a2)n 



dg 



g'"(g2-a2)"-i 



66 



INDEFINITE INTEGRALS 




14.163 
14.164 
14.165 
14.166 
14.167 
14.168 
14.169 
14.170 
14.171 
14.172 
14.173 
14.174 
14.175 
14.176 
14.177 
14.178 
14.179 

14.180 
14.181 



J a^ — x^ 2a \a — 

J a2- 



or — tanh~ ' — 

X 



dx , a, I a + X 

x^ 2 \ a — X 



c^ dx 



2 2 



j^^ = -^-~}n(a^-x^ 



f dx ^ J_ / gg 

J x(a2 - a;2) 2a2 '" \^a2 - 

f dx ^ _J^ , J^, /« + 

J xHa2 - x2) a2x 2o3 *" \^a - 

f '^a: L- , J_ 1 f-^Lj] 

J x3(a2-x2) 2a2:c2 ^ 20" \a2 - ara^ 

r dx ^ _ 

J (a2-x2)2 2a2(a2-x2) ' Aa^ "\a - x 

J x dx _ 1 
{a^-x 

J (^ 



(a2 - a;2)Z 2(o2 - x2) 



2dx 



_ 1 l„/a+x 



x2)2 2(a2 - x2) 4a V^a - x 

J (a2 - -c2)2 2(a2 - x2) ^ 2 ^ ' 



+ :=^ln 



r dx_ 

J x(a2 - x2)2 2a2(a2 - x2) ' 2a'' "* \^a2 - x^ 

/ dx _ — 1 , X , _3_ , I a + X 

xHa^ - x2)2 a*x 2aHa^ - x2) 4a5 \a - x 

r ^g ^ -1 , 1 + J_ In / g^ 

J x3(a.2 - x2)2 2a4x2 aa^ias - a;2) o* \a? - x2 

f dx ^ X 2n - 3 r dx 

J (a2-a;2)n 2(n- I)a2(a2 - x2)r.-i "^ {2n - 2)a2 J (a2_a;2)n-i 

J xdx 
{a2 - X 



2)« 2(tt-l)(a2-x2) 



/ 



dx 



1 , i_ r dx 

,-l)a2{a2 — a:2)n-» a2 J a:(a2-x2)"-i 



x(a2 - x^)" 2(K 

(a2-x2)n "• J (a2_3;2)n J (a^ - x2)n-\ : ■'< - 

f dx ^ 1_ r dx , J_ C dx 

J «™(a2-a;2)« a2 J a'n(a2- a:2)n-i "•" a? J 3:^-2(^2 _ a-2)n 



■»- . r 



r ' r> 



1 



INDEFINITE INTEGRALS 



67 



INTEGRALS INVOLVING V*^Ta^ 



14.182 
14.183 

14.184 
14.185 

14.186 

14.187 

14.188 

14.189 
14.190 
14.191 
14.192 

14.193 

14.194 

14.195 
14.196 

14.197 

14.198 

14.199 

14.200 

14.201 
14.202 



dx 



f . ""^ = In (a: + VicM^ ) or sinh " ^ - 



cV«^ + o,^ a2 



^ V^MT^ 3 

xV'x2 + o2 



ln(x + VkM^) 



1 j a + y/x^ + d^ 

= - - In 

a \ X 



/- dx ^ Va=2 + a2 

J a;2^a;2 + a2 



a^x 



Va;2 + o2 1 /a + \/x2Ta2 

+ ^;-:i In 



2a2a:2 2a3 

3;V^2 + a5 0^2 



+ ^ In (X + \/x2 + ^2 ) 



/ da: _ 

x3\/x2 + a2 

I Vx2 + a2 dx = 

rx2,/^2T^dx = .(x2 + a2)s;. _ g^vgrg _ 4 ^^ ^^ ^ ^^-.-^^ 
v' 4 8° 

fxaV^T^d^ = (x2 + a2)5/3^^2(:,2 + ^2)3/2 

./ 5 3 

r y/x^ + a2 , „ „ ^ fa + y/x^ + a'^ 

J a 



dx = y/x^ + a2 — a In 



Vx^ + a2 



J 

./ X 



dx - - 



\/x2 + a2 



+ !n (x + ^3:2 + a2 ) 



Vx2 + a2 



dx = — 



\/x2 + a2 1 /a + \/x2+a 



2x2 



--^In 



J (x2 + a: 



X dx —1 



f x^dx _ —X 

(x2 + a2)3/2 - .^/^5T^ + ln(x + -/^24^) 

J (x2 + a2)3/2 ^^2 + a2 

J x{x2 + a2 



1 /a + Vi^Ta2^ 
;;ln 



r dx ^ V^' + "■' X 

J x2(3;2 + a2)3/2 a^X ^4y^2+^ 

r dx -1 3 

J x3{x2 + o2)3/2 



fa + y/^ + a2\ 



202x2^ + 02 2a4Vx2 + a2 



+ 2!b1" 



a + Vx2 + a2 



68 



INDEFINITE INTEGRALS 



14.203 f (a:^ + a2)3/2 da: = 

14.204 r x(a;2 + a2)3/2 da; 

14.205 ra:2(x2 + a2)3/2da: 

14.206 r a:3(3;2 + a2)3/2 da; 

,4.207 I i:?i±^>^.x = 






_ (^2 + a2)5/2 



x(x2 + a2)5/2 a2a;{x2 + a2)3/2 a^^V^M^ „6 . , „ , 

- 6 24 16 - jglnCt + V^^T^) 

_ {a:2 + a2)?/2 (l2(a;2 + ^2)5/2 



J 



(a;2 + a2)3/2 



+ a2V»2 + a2 - a3 In 



a + V^2T^ 



(x2 + a2)3/2 



14.208 1 '-^ : ^ dx = 

a;2 



,4.209 I "'^ + f'"' d. = 



(3;2 + ^2)3/2 3x-\/xM=^ g 



2 + 2a2 1n(x + VxM^) 



/ 



2:r2 +g/x2+a2--aln 



INTEGRALS INVOLVING \/a;2 - a^ 



14.210 

14.211 
14.212 
14.213 

14.214 

14.215 

14.216 
14.217 

14.218 
14.219 

14.220 

14.221 

14.222 
14.223 



r^^^__ ^ In {x + y/x^ - a2 ), C -^^^ ^ V^^^^- 

J ^a:2 - a2 J V^sTT^ 

f_x2d^_ x\/a;2-o2 ^g ^ ^ 



dx 



= — sec~^ 



xy/x^ - a2 « 



Va:2-a2 ^ 

xyx'^ — a2 ^2 



J 

r 

•^ x^yjx"^ — a?- 
C dx 
•^ a; Va:- - a^ 

r Va:" - a2 dx = ^ 

jx^f^^^dx = M^!)!^ 

I x^yj x^ — d 

.7 5 3 

C Vx2 - a2 

J ^2 

C dx 

J (a:2-a2)3/2 



-^\n{x-¥ y/x^-a^) 



a;(a;2 - ^2)3/2 a^x^f x"^ ~ a^ a*, / ^ r^ 2\ 

2 da; = — ^ ;; 1 r- In (a; + \x^ — a^ ) 



dx = -\/x^ — a2 — a sec~' — 

I a I 

Vx^ - o2 

dx = + In (a; + Va;2 - a^ ) 



-\/x^ — a2 J 



2a:2 

X 



2a I a I 



a2-/a:2 - a2 



INDEFINITE INTEGRALS 



69 



14.224 
14.225 
14.226 
14.227 

14.228 
14.229 

14.230 
14.231 

14.232 
14.233 
14.234 

14.235 

14.236 



J x dx _ —1 

r dx 

C dx 

} a:2(a;2 - a2)3/2 

r dx 



- -3 sec 1 



Vx^ - gg _ a; 

1 3 



)3/2 



J(.-„.j3.,, . x(.2 - „2)3. _ Sa^xV^ ^ 1^, ,^ ^^ ^ ^.^.^^ 



fxHx2-a^)^ndx = (a:^-a2F^^ a2(,2-„2)5/2 
t/ 7 5 

•^ ^ 3 I a I 

r (x2 - a2) 3/2 
J X^ 



(a;2_ 0.2)3/2 3a:\/a;2 - a2 3 



-■|a2 1n{x + Va^ - a2 ) 



(g2 _ q2)3/2 3^x2 - gg _ 3 

2x2 "^ 2 2 



INTEGRALS INVOLVING V^^TT^ 



14.237 r , '^^ = sin-i^ 



14.238 



J v/q,2 „ 3.2 



14.240 
14.241 
14.242 
14.243 



/~2 j_ x-Ja^ — x^ „2 

Va" - a;2 2 2a 

r ^'^tfx _ {a2 _ ^2)3/2 — 5 

r rfx ^ _]_ / g + Va2_a.2 

J xVa2 - x2 '^ " V 3: 



Va^~x2 



J 3;Z^tt2 _ a.2 a^x 

X rfx 
x3Va2 — x2 



Vo^ - x2 1 /a + Va^-x 



7^ In 



2a23;2 2a3 



fr 

r 



70 INDEFINITE INTEGRALS 



r Va2 - x^ 



14.244 I yfd^^x^dx = ^^^^— ^ + f sin-'l 



14.245 I x^a^-x^dx = _ (5i__^!l!^ 



I xy/a~ — x^ 



14.246 J a;2^a2-x2dx = _^i^_-!_ + v_ +__sj„-i_ 



14.247 



14.248 



J O 



14.249 I- — s dx = sin-'T 



14.250 J ^^ — ^ — rfa: = - n^g + ;r: 1° ' ^ 



2ic2 ^ 2a 



^^•^5^ /(^ '^ 



2^x2)3/2 „2V^rr:i^ 



14.252 Tt^'^'^ 



)^'^ V^ 



14.253 I Y^y 2^372" = , - s'n^- 

,4.254 f-V^i^ = V^^:^+^^= 

/dx 11, f g + Va^ — a?2 '\ 



p dx _ -1 3 3_ J / tt + Va 

14.257 J ^3(„2_«;2)3/2 - 2a2x2V^3^^ ^ 2a^^^^^^ ^a^ "V 



14.258 J (a2-x2)3/2da: = ^ ^ ^ + ^^8 + ga* sin »- 

J/-,2 _ 3:2^5/2 
X(a2-x2)3/2dx = -IB-^J— 



,4.260 J .^(.-.=)«.. = -2(2!:^ + -'-<"'--''"' ^ ^^^f^ + fi^in-.f 
14.261 /.»(.-x=)«<£x = i^-=fg^- °^'°^ --''"' 

,4.264 Ji °^-f'" .. = _(£!^^_?VHEE?+|„„(£±3^) 



INDEFINITE INTEGRALS 



71 




14.265 



J 



dx 



ax^ -f 6a; + c 



2 , 2aa- + & 
tan~' 



V4ac — 62 yf^a^^^^ 



In ' 



V62 - 4ac \ 2aa; + 6 + \/b^ - 4ac , 



If 62 = Aae, ax^+hx-¥c = a{x + 6/2a)2 and the results on pages 60-61 can be used. If 6 - use 
results on page 64. If a or c =^ use results on pages 60-61. 



14.266 



dx 



ax~ + hx ■¥ c 



'^"''^ J aa;2+6x + c (jn-l)a a J ax^ + 6x + c a J 



dx 



ax^ 4- 6x + c 
x""^^ dx 



14.269 



J 



dx 



= i^In 



14.270 



x(fflx2 + 6x + c) 2c \^ox2 + 6x -I- c 

dx 6 



2c J 



ax2 + 6x + c 
dx 



ax^ + 6x -I- c 



KL 



dx 



az2 + ba; + c 



C dx _ _6_, / ax2 + &x + c N _ J_ . ^^ - ^ac T 

J x2(ax2 + 6x + c) 2c2 \ x^ ) ex %<^ J 

,.„, f dx ^ _ 1 b f dx ^ f — 

J x"(ax2 4- bx + c) (n-l)cx"-i cj x'»->(aa;2 + bx + c) cj x" 

Id 979 r tfx ^ 2ax + 6 , 2a. f 

i*i.i/x J (^3.2 + ^3.4.^)2 (4ac - b2)(ax2 + 6x + c) 4ac - b\} 



dx ' 



-2{ax2 + 6x + c) 



dx 



14.273 
14.274 
14.275 



/ 



(ax2 + bx + c)2 (4ac - b2)(ax2 + 6x + c) 4ac - b^ J ax^ + 6x + c 

xdx __ bx + 2c b r dx 

(ax2 + bx + c)2 

2dx 



. to 



S • - 



-bsj 



r ^ 

J (ax2 + 

J (ax2 + bx + c)" 



{4ac - b2)(ax2 + fcx + c) 4ac 
(b2-2ac)x + be + 2c T 



aic2 + bx + C 
dx 



+ bx + c)2 a(4ac - b2)(ax2 + bx + c) 4ac - b^ J ax^ + bx + c 

mdx -''"-' (m— l)c i x'"~2dx 



(2n - m - l)a(ax2 + bx + c) 

(n — wt)b C x"*~* dx 



(m-l)c f 

"-I (2n — m-l)aj 



(ox2 + 6x4- c)" 



.X 



14.276 

14.277 . j(^2 _j_ bx + c)2 ~ 2c(ox2 + bx + c) 

14 978 r ^^ ^ 1 _3g r 

1.^/ o J ^2(a^2 + 6a; + c)2 cx(ax2 + 6x 4- c^ c J 

14.279 f 



{2n-m-l)aJ {ax2 + bx + c)" 

2'>-3 dx _ _ ^ f 



x2''-2 dx 



X x^^-^ dx _ 1 C x2"-3 dx _ c r X 

(ax2+bx + c)'' ~ a) {ax2 + 6x + c)"-i a J (ax2 

c dx ^ 1 _ J. r '^'^ I ^ f 

J x(ax2+bx + c)2 2c(ox2+bx + c) 2c J (aa;2+6x + c)2 cj x(ax2 4-bx + c) 



+ bx 4- c)n a J (ax2 + bx 4- c)" 
dx . 1 r dx 



dx 



c2(ax2 + bx + c)2 
dx 



cx(ax2 + 6x 4- c) 
1 



(Ox2 + bx + c)2 



_ 2b f 

C J X 



dx 



(ax2 + bx + c)2 



(m + 2to — 3) 



x'"(ax2 + bx + c)" (wi - l)cx"'~Hax^ + bx + c)"-i (m- l)c 

dx 






dx 



2(aar2 + bx 4- c)" 



(m + n — 2)b 
(m — l)c 



J ar"*-' 



(ax2 + bx 4- c)" 



INDEFINITE INTEGRALS 



INTEGRALS INVOtVING \/a^Tb^Tc 

In the following resulte if 62 = 4ac, Vax^ + bx + c = yfa{x^ 6/2a) and the results on pages 60-61 can 
be used. If 6 = use the results on pages 67-70. If a = or c - use the results on pages 61-62. 



14.280 



/ 



dx 



14.281 r 

14.282 r 



yjax"^ + hx + c 



xdx 
■\/ax^ -f- 6a; -f c 

x^dx 
y/ax^ +bx + c 



~^\Tl{2^/a^J ax^ + bx + c + 2ax + 6) 
y/a 

1 .„^i / 2ax-¥ b 
L V-a \yt2-4acj V^ 

yjax^ -\-bx + c h 



or sinh 1 



y/iac — b^ 



2oJ 



dx 



y/ax^ + bx + e 
_ 2ax - 36 j „ 2 , , , , 362- ^^c f 



da; 



Voic^ -i- 6a; -|- c 



14.283 



14.284 



14.285 



/ 



dx 
xy/ax- -I- 6a; + c 



dx 



_ J_ In ( 2y/cy/ax^ +bx + c + bx + 2c 
yfc 



sin' 



6a; + 2c 



■■^y/ax^+JxTZ 



I \^ax^ + bx + c dx = 



y~^ \\x\yjb^-4ac 

_ y/ax^ +bx + c b 



2c J ^ 



1 • U-. / bx + 2c 
or — -— sinh ' / 

Vc yx\y/4ac-b^ 

dx 



y/ax'^ + bx + c 



{Zax + 6) Va«2+"6xTc 4ac — 6^ 



4a 



+ 



8a 



/ 



dx 



y/ax^ + 6a: -f c 

14.286 rxVax2 + 6a: + cda: = (ax^ + 6x -H ^)3/2 _ b{2ax + 6) . ^^, ^ ^^ ^ ^ 

_ 6(4ac - 62) r da; 

1^«' J y/ax^+bx + c 

14.287 JWart^-hfrx + cdx = ^"'^'J^ ^^^^ + ^'^ + '^>''^^ + ^^'leJ^^ ' f V «=^^ + 6a; + c da; 

dz 



14.288 



14.289 



/y/ax^ + bx + c , , h f 

^ dx = yjax"^ + 6a; -I- c -h | j 



di 



-I- c 



J 



Vaa;2 + &g-l-c , _ yfa^^^hxVc 



yjax^ -H 6a; -I- c "^ xyfaifi^Vbx + c 



•^ a;i 



da; — — 



14.290 /^ 

14.291 r 



dx 



2(2ax -!- 6) 



+ 



-/ 



da; 



\/a«2 -i- 6a; -f- c ^ -^ xy^zM^TxTc 



/ 



dx 



ax2 + 6x -H c)3/2 ^^^^ _ j^,^ y^2+6^ + ^ 

xdz 2(6a; + 2c) 



14 292 r ^^'^'^ :^ (26^-4ac)a; + 26c ]. r da; 

J („^2 + bx -K c)3/2 ^(4^^ _ ^,^ Vaa;2+6x + c "^ a J ^ ^^2 + 5^ + , 

14.293 



J x( 



dx 



-I- 



^ (62 — 4ac) vtta;2 -I- 6x -I- c 

_ 1 r dx b_ r dx 

iax^+bx + cY'^ ^^^2 + 6^ + ^ "^ cj ^^/^^FT6^T. ^cj (ax^ + 6a; 

14294 f ^^^ - o3;2 + 26x -t- c 62 - 2ac r dx 

J x2(ax2+6x-Fc)3/2 fl23:^ax2-K6x-Fc 2c2 J (a;^2 + 6^,4^)3/2 

36 r dx 

^''^'^ x\/ax2T6xTc 

14.295 ( {ax-^-^bx + c)^ + ^'^dx = (2ax + 6)(gx2 + 6x-Hc)"+i/2 
-^ 4a(m + 1) 



+ c)3/2 



INDEFINITE INTEGRALS 



73 



14.296 
14.297 

14.298 



J, 



2(2ax + 6) 



(0x2 + 6a; + cY + 1'2 (2m - l)(4ac - 62)(ax2 + 6a: + cY~^'^ 

8o(n — 1) r da: 



+ 



/. 



dx 



(2w-l)(4ac 
1 



-62) J (aa: 



(ax2 + 6x + c)" + >'2 (2n - l)c(aa;2 + 6x + c^-^i^ 

dx 



+ 



'J 



x(ax2+6a; + c)"-i/2 



2 + 6x + c)"-i'2 



2c J 



d:e 



(oar2 + 6x + c)«+i/2 



INTEGRALS INVOLVING x^ + a^ 



Note that for formulas involving x^ — a^ replace a by —a. 

,4.299 f^^ = ^,„_i£±^ + ^tan-^^ 

1 , a:2 _ aa, + a2 j _, 2x-a 

■T— In — -. — ; — 7K 1 — z tan ' p— 

6a (x + a)2 ^^ „^ 

14.301 (' ff^, = |ln(a:3 + a3) 
J 3;3 + a3 3 



14.300 



/ xdx _ 
a;3 + a3 



dx 



ay o u-v " 

;.3 + „3) ,4.302 /^(^ = 3^3ln(^ 

j_ _ j_ x2 - Qx + g'i _ ^_ ^^^„, 2x^1^ 
a3x 6a4'" (x + a)2 ^,^ ^^3 



ax + a.2 
a:2 — fflx + g2 



3a4\/3 



14.303 J a:2(x3+a3) ^03; oa- vx -r or 

wsAA r_-^5— - g , J_, (g + «)^ 

14.304 J (j.3 + a3)2 - 3a3(x3 + a3) "^ 9a5 ^" x2 - ax + 

lAtn^ r ardx _ a:2 , 1 .^ a:2^axjt^ 

I4.J03 J (a:3 + a3)2 " 3a3(x3 + a3) ^ 18a* (x + a)2 

,--*- r x2dx _ 1 

J (x3 + a3)2 3{x3 + a3) 

^^•^°^ J x(x3 + a3)2 ^ 3a3(x3 + a3) "*■ 3^ ^" Vx^ + a3 j 
I^.JUlf J ^2(a^ + q3)2 a^x 3a«(x3 + a^) Sa^ J x^ + a- 

14.309 J ^^.p^ = ^^ - "^ J -^T^ 

J^ —\\ C dx 

x"(x3 + a3) = a3(n-l)xn-i " a V x"-3(x3 + aS) 

INTEGRALS INVoK^^^^^C? 



^ , 2x — a 

-^tan-i 3- 

Sa^VS aV3 

2x - 



+ tan"! 



[See 14.300] 



14.311 
14.312 

14.313 

14.314 



x2 + axV2 + o2 
a^ + a* ~ 4a3\/2 "' V*^ " «*V2 + ©2 



X 

C xdx 

} x^->r a'^ 

C x^dx 
J X* + a* 

J x^dx 
x^ + a* 



In 



= ::-;; tan » -5- 



2a2 



1 /x2 — 0x^2 + a2 

In 



4av^ \a2 + ax\/2 + a^ 



1 . _. axv2 
-tan 1 „ „ 

2a3V2 * " '^ 



^ tan- ^1^ 
2ay2 a:2 - a^ 



J In (x* + a*) 



74 



INDEFINITE INTEGRALS 



14.315 



/ 



dx 



^ T^ In 



x(x'* + a") 4a4 \x^ + a^ 



2aG^""'f^ 



14317 r ^'^ - ^_}: 1 

14.318 f-r^ = J_ln/^^^VA 

14.3.9 J,!^ = i^I-nfe 

M.3ao J,5!i?|^ = -L,„ 



^ „, axy/ 2 

^ tan ' -s 5 

V2 * ~ ** 



tan-i- 
a 



z — a\ , 1 . _, X 
— ; — 4- ;;- tan 1 — 
x + aj 2a a 



14.324 r 



dx 



1 , 1 , / 3:2 - a2 
+ -r^ In 



x3(3ri - a*) 2a'»x2 4a6 y ^2 4. ^z 



INTEGRALS INVOLVING a;" ± a" 



14.325 

14.326 

14.327 

14.328 

14.329 

14.330 

14.331 

14.332 

14.333 

14.334 



r dx_ 

J x{x^ + 



+ a") 



^ In ^^ 



na'" a;" + a" 



fx"^*dx 1 



/pwi— n (£2; 



/ x"'dx _ r x"*~"da: _ „ T 
(x" + a")' ~ J (x" + a")'-! "■" J (x" + a")*- 

X dx _ J_ r dx J_ r <ix 

x'"(x" + a")'' a*^} x'"{3;" + a")''-* a" J 



dx 



x"*'~"{x" + a^y 



_ 1 / ^/x" + g" - \/a" \ 



xV^*^" + a" mV'a" \\/x*» + a" 4- yf^J 

/ dx _ _1_ - / g" — a" \ 

x(x" — g") Tia" I X** / 

Jxn-irfx 1, , 

—;, ;: = - In (x" — a") 

X" — a" n ^ ' 

/ x^dx _ ^ r x"'-"<?x r x""" dx 

{x^'-a^y * J (x"-g")'' J (x''-a")'— 1 

r dx ^ 1 r dx ?^ f — 

J dx 



da; 



2 . / o" 

:cos^ 



INDEFINITE INTEGRALS 



75 



14.335 






2m 



,2m -p 



(2fc - l)pir / a: + a COS [(2fc - l)W2m] 

2 sin — s — tan^i 



fc=i 



a sin [{2k - l)jr/2m] 



y 



where < p ^ 2m. 



2ma2m 



1.2m -p j^^j^ 2m \ 2m J 



14.336 






a2m 



fepr 
m 



k^ 



2 cos— ^ In [ a;2 — 2(ixcos-^ + a 



14.337 



14.338 



1 '"■^' . fcprr . _, /a: — a cos (fc^/m) 

o„-n 2 sin -^— tan ^ : — ., , -: — 

^a2m-p ^ ^ 1 a sin (fcirArt) 



where < p ^ 2m. 



J x2m + 1 4- a2m + 1 



+ 2^i^t^°^^"''^ + (-l)Mn(x + a)} 



2(_i)P-i ™ . 2fcp^ , _i / x + g COS [2kirl{2m + 1)] 

(2m+l)a2--P + ifc^i "'" 2m + 1 ^^" 



(_l)P-i 



2kpir 



asin[2fc7r/(2m+l)] 
2fe7r 



-^ 2 cos^^^ In x2 + 2ax cos^^^ + a^ 

(2m+l)a2'"-p+ifc=i 2m+l \^ 2m + 1 

(-l)p-Mn(x + a) 

(2m+l)a2'n-p+» 

where < p ^ 2m + 1. 

j.2m + l _ (i2m+l 

- -2 ^ _in ^'^/^ tan-i f ^^ 

(2m+ l)a^'n-P+i ^-f , ="" 2m + 1 1, a sin [2K/(2m + 1)] 



_2 ■'• ZkpTT - /''^ ~ a cos [2fcjr/(2m + 1)} 



+ 



(2m + 1) 



1 ^ 2fcps- , / 5 „ 2fc:r , „ 

^ TT 2 cos -= — —■ In a2 _ 2ax cos ■= t—t + or 

j(j2m-p+i^"^j 2m + 1 Y 2m +1 



+ 



In {x — a) 



(2m + l)a2m-p + i 



where < p ^ 2m + 1. 



INTEGRALS INVOLVING sin ax 






i> ■ 



14.339 
14^40 
14.341 






sin ax dx — — 
sin ax dx = 



cos gjc 
a 

sinour 
a2 



X cos ox 



2x f 2 x^ 

x^ aiaax dx = -j sin ax +( -3 — — J cos ax 



14.342 (n^ainaxdx = (^ " ^) s'"*^ + fe" ^ 



14.343 
14.344 
14.345 
14.346 
14.347 



/^ 



sm ax , (ax)3 , (ax)^ 

dx = ax — ' '. + 



/ sin. 
a:' 

J sin 
J sir 



ax 



dx = - 



3-3! 
sin ax 



+ 



X 



»J 



5'5! 
cos ax 



X 



dx [see 14.373] 



ax 
dx 



— — In (esc ax — cot ox) = - In tan -^ 
a a £. 



1 j , (ax)3 , 7(ax)5 



2(22n-l - l)B„(ax)2n+l 

■^ (2n+l)! ^ 



sin^ ax dx — -^^ — 



X sin 2ax 



4a 



^ M 



^': 



-S. 



INDEFINITE INTEGRALS 



14.348 
14.349 
14.350 
14.351 
14.352 
14.353 
14.354 
14.355 
14.356 

14.357 

14.358 
14.359 

14.360 



X sin2 ax dx = ^ - ^ ^^" ^"^ _ ^os ^aa; 
4 ia 8a2 

sm^axdx = -^osax ^co^ax 
a 3a 

sin^ axdx - ^ - s'" ^"^ , sin Aax 
8 4a ^ 32a 

dx 



/ 
/ 
/ 
J 
.f 
J 

J 1 — sin 

J 

J 



sin^ ax 

rfa; 
sin^ax 



— cot ax 

a 



2a sin2 aa: 2a 



+ t:- In tan -r- 



sin P.C sin 9X dx = ^'" ^P " g>^ _ S'" ^P + ?)» 
2(p-9) 2(p-h9) 



[If p = ±q, see 14.368.] 



^ ltan(f+^ 
ax a V4 2 



i — sin ax a \ 4 2 



4lnsin -?- 



dx ^ -ltan(^-°^ 

1 4- sin ax o \ 4 2 



J 1 + sin 



{X — sin ax)2 



r ^ 

J {X - si 



= -^tan(^-«-^ 
sin ax a V 4 2 



dx 



_i_ ^ 1 . / f ox 
H s In sin T + ^r- 



„ -tan ^ + ^ 
2a \ 4 2 



+ f- tan3 [ 5 + ^ 
6a \ 4 2 



(l-sinax)2 - "2^*^"(4"'2";- 6^**"H4-T 



J 



dx 



p -f g sin ax 



a-y/^2Tr^ 



tan-i 



p tan ^ax + q 



Vp2l-g2 



In 



p tan iax + 9 — vg^ — p2 



14.361 

14.362 

14.363 

14.364 
14.365 
14.366 
14.367 
14.368 



If p= ±q see 14.354 and 14.356. 

(?x o cos ax . JJ i^' dx 



J 



Q COS ax 



+ 



(p + 9 sin ax)2 a(p3 - g2)(p ^ ^ gin ax) P^ 

If p = ^q see X4.358 and 14.359. 






+ q sin aa: 



Vp^ + 



\/p2 _ q2 tan 



/dx 1.1 y'p2 4- q,2 tan ax 

/ 

f^rmsinaxdx = ^ ^^ cos ax ^ mx-"-isinax _ m(m-l) C ^rn-2 ^j^^^ g^ 

J a a^ a^ J 

rsinox^^ = _ sinaa^ _^_ _g_ T gosox ^^ [see 14.395] 

f sin" axdx = _sin"-iaxcosax ^ »^i_l T ^.^„_, 

J an n J 



p^ + q'^ sin2 ax 



p^ — (p- sin2 ax 



Vp^^2 P 

^(72 _ p2 t;an ax + p 
lapyjrp- — p2 Vvg^ — p^ tan ox — p 



' 2 ax dx 



/ dx _ — cos gx , n — 2 ^ dx 

sin" ax a(n — 1) sin"~* ax n— 1 J sin""^^^^. 

J xdx _ — z cos ax 1 w — 2 f" x dx 

sin"ax a(n — 1) sin""" ^ ax a2(w,- l)(n — 2) sin"-2ax n — 1 J sin"^^ 



INDEFINITE INTEGRALS 



77 



^OTEGRAIS rNVOlVlWs''^^^^ 



14.369 

14.370 

14.371 

14.372 

14.373 

14.374 

14.375 

14.376 

14.377 

14.378 

14.379 

14.380 

14.381 

14.382 

14.383 

14.384 

14.385 

14.386 

14.387 

14.368 

14.389 



/, _ sin ox 
cos ax dx — 
a 

/, _ cos ax , X sin ax 
X cos ax ax — — + 
a^ a 



;2 COS ax dx — —^ COS ax + [^ — -^ ) sin ax 



a a- 



jx^cwaxdx = (^-±jcosax + [^-—Jsmax 

Xcosax . , {ax)2 (ax)* {ax)'^ 4. . . . 

CS21^dx = -S£l^^ a (^^^^^dx [See 14.3431 

J X^ X J X 



= "In (sec ax + tan ax) = — In tan t + "^ 



COS ax 



J 



xdx ^ J_ Uax)^ (ax)^ 5(oa;)a 



cos ax 



+i5r+ 



"^ (2n + 2)(27i)! ■*" 



/„ , a; , sin 2aa; 

cos^ ax dx = -x + 



4a 



J- _, x^ , ;c sin 2ax , cos 2ax 

C , , sin ax sin^ ax 

I cos3 ax dx — r 

J a 3a 

JZx sin 2ox , sin iax 

cos* ax dx - ^ + -^^ + -^2^ 

J 



_ tan ax 



cos^ ax 
dx 



r_^ = sinax + 1 intan('^ + ^ 
-/ cos3 ax 2a cos^ ax 2a \4 2 

J, sin (a - p)x , sin ( a + p)x 
cos ax cos px dx = ^ : f- 



2(a-p) 



2{o + p) 



Xr^ 



dx 



1 .ax 

= cot -r- 

cos ax a ^ 



h 
St 



dx 



cos ax 
dx 



X . ax , 2 , . ax 
- cot -r- + -5 In sin -^ 
a 2 a^ 2 



1 ^ ax 
- - tan -^ 
+ cos ax a 2 



Jo- 
/ 



_JEA^_ ^ ^tan^+ ^.Incosf 
+ cos ax a 2 a^ £ 



dx 1 ax 1 3 ax 



dx 



(1 + cos ax)^ 



1 ^ ax , 1 . „ ax 
■^ tan -^ + -F- tan-f -^ 
2a 2 6a 2 



[If a ^ ±p, see 14.377. 



78 



INDEFINITE INTEGRALS 



14.390 

14.391 
14.392 

14.393 

14.394 
14.395 
14.396 
14.397 
14.398 



h 



dx 



q cos ax 



a\p^ — 



. tan- ^ ^/ (p - q)/{p + g) tan^a; 



In 



tan ^ax + v'(g'+p)7(9--p) 



[If p = :tg see 
14.384 and 14.386.1 



dx 



J {p + q cos ax)^ 
dx 



ay/q^ — p2 ^tan ^ax — y/{q + p)li.q — p)l 

q sin ax _ p C dx 



J 



a(g2 _ p2J(p ^ q, (.pg g^jg^ 



, p tan ox 
tan"' 



+ g cos ax 



[If p = ±g see 
14.388 and 14.389. 



p2 + g2 cos2 aa: apy/^^T^^ y/p^ + q^ 

J p^ — q2 cos^ ax 



dx 



1 . 1 P tan ax 
■ — tan~i ~ 

apVp^ — q^ yjp^ ~- g2 



p tan ax — yjq^ — p^ 
2ap-\/q^ — p2 \p tan ox + ^/q^ — p^ 



In 



x-" cos ax dx = = — »ui_ux ^ »«x — cos ax s — ■ I x"" ^ cos ax dx 

a a^ o^ ./ _ 

Tcos^ ^ _^osax _^ C^^dx [Seel4.365] 



fcosnaxdx = sin ax COS"-' ax ^ ZL^li C 
J an tf- J 



r do 

J cos" 



sm ax 



ax 
xdx 



a(n — 1) cos' 

X sin ax 



^ « - 2 r da 

"^ ax n — 1 ^ cos"- 



cos""2aa! dx 



dx 



cos" ax a(re - i) cos"" ' ax aHn — l)(n — 2) cos''-^ oa; 



+ 



K-2 r xd 

n — 1 J cos"*- 



dx 

2 ax 



INTEGRAIS INVOLVrNG sin ax AND cos ax 



14.399 
14.400 
14.401 
14.402 



az cos ax dx = 



J sin 

fsinpxcosgxdx = - COs(p-q)x _ cos {p + q)x 
J 2{p - q) 2(p + g) 

I sin" 
I cos" 



ax cos ax dx = 



2a 



sin" * ^ ax 

{n + l)a 



[If n = -1, see 14.440.] 



ax sin ax dx = — 



cQs"'^^ ax 

(n+ l)a 



[If n = -1. see 14.429.] 



1 4.403 1 sin^ ax cos^ ax dx = — — 



14.404 f- — 

J sm a 



8 32a 



dx 



14.405 f^- 

J sm 

14.406 r^- 

J sm 



X cos ax a 

dx 



= - In tan ax 



1 , ^ / V , ax \ 1 

= - In tan t + -h" ~ r 



2 ax cos ax a \i 2 / a sin ax 

dx 



1 . ^ ax , 1 

= i In tan -r- + 



14.407 



/ii 



ax cos^ ox a 2 a cos ax 

dx 2 cot2ax 



sin^ ax cos^ ax 



a 



>-^ 



INDEFINITE INTEGRALS 



79 



14.408 
14.409 
14.410 
14.411 
14.412 
14.413 
14.414 
14.415 
14.416 
14.417 

14.418 
14.419 

14.420 






COS ax 



dx = - 



, 1 , , I ax , IT 



J cos 
sii 

I 



dx ~ 



dx 



a a 2 



cos ax(l ± sin ax) 

dx 

sinofl:{l ± cos ax) 

r dx 

.) sin ax ± cos ax 

ax dx 



f sin aa 
sin ax ± 

J 



cos ax 
cos ax dx 



1 + f lntan(^+J 

2a{l ± sin ox) 2a \2 4 

1 , 1 , . ax 

+ ;;- In tan -r- 

2a(l±cosax> 2a 2 



av^ V2 8 



X 1 , , . j_ . 

-^ ^ ^ In (sin ax ± cos ax) 



sin ax =: cos ax 
sin ax dx 



X 1 
= ±— + — In (sin ax ± cos ax) 



J p4 



I = In {p + g cos ax) 

+ q cos ax aq ' 



cos ox dx 



J p + g sin ox aq 

C sin ax dx 



In (p + g sin ax) 
1 



(p + g cos ax)" aqi^n — l)(p + g cos Ox)»~ ' 

C COS ax rfx _ —1 



(p + g sin oz)" ag(m - l)(p + g sin ax)"-i 



.7 p sin 
./ p si 



dx 



ax+ g cos ax a^jrP■+q^ 



In tan 



ox + tan ' (g/p) \ 



dx 



sin ox + g cos ax + r 



a-\]i^ ~ p2 - g2 



tan" 



^ / p + (r — g) tan (ax/2) 



In 



p _ Vp2 + q2 _ r2 + (r - g) tan (ax/2) \ 
a\/p2 + ^2 _ r2 "" Vp + Vp2 + g2 - r2 4- (r - g) tan (ox/2) / 
If r = g see 14.421. If r2 = p2 -j- ,2 ggg 14.422. 



14.421 



14.422 



14.423 



14.424 



14.425 



/ 



dx 1 , / , , ax 

■• T — TT-T T = — In g + p tan ^ 

p sin ox + g(l + cos ax) a-p \ 2 



dx 



p sin ax ■\- q cos ax ± vp^ + g* aVp^ + g^ V 



-1 ^J.^ ax^tan-Mg/p) \ 



J p- SI 



dx 



_ 1 , _.^ I p tan ox 

sin^ ox + g2 cos^ ax apq V g 



r dx 

J p^ sin^ ox — g^ 



In 



cos- ax 2opg \ p tan ax + q 



p tan ax — q 



J== 



sin"" ax cos" ox dx — 



sin"* ' ^ ax cos" "^ ' ax , m 



a{m+n) m + 

sin"" + ^ ax cos" "" ' ax 



o(m + n) 



+ 



/3in"''"2 ax cos" ax dx 

Jsin"^ ax cob''"^ qx dx 



80 



INDEFINITE INTEGRALS 



J C( 



14.426 I ?H^^dx 



cos" ax 



a(n 



sin'"~^aa; _ m — 1 C sin*" ^ ax 

— l)cos"~^aa; n — lj cos''~^c[x 



(i(re — 1) cos 



— sin"* ^ aa; 
a{m — n) cos"~i oa; 



1 gg _ wi — n + 2 r _^ 

)s"~^ ax n — 1 J cos 



sin"" ax 



co&"^^ax 



dx 



+ 



m — 1 I s]n"'~"2 ax 



cos" ax 



- 1 Tsi 

— M J I 



da; 



14.427 



/CDS'" aa; , 
^ *^ac = 
sin" ax 



— CDS'"" ^ ax _ m — 1 r cQS*"~^ ax 
a{n — 1) sin"~i ax n — 1 J sin^~^ ax 

— cos'" + ^ax m — 71 + 2 C cos'" ax 

a(m— 1) sin"-' ax n-1 J sin"-2ax * 

— 1 r cos" 

— n J sii 



COS"" ' ax 



,a(m — m) sin"- 'ax wt 



cos"* -^ ax 
sin" ox 



14.428 



/ii 



dx 



sin"" ax COS" ax 



J^ ^ m + 7i- 2 f 

"• ax cos"-' ax n — 1 J si 



dx 



a{n — 1) sin" 

^1 _j_ m + ti-2 r 

a(«i — 1) sin™-' ax cos"-' ox m— 1 J sin'"- 



sin"" ax cos"--' ax 



dx 



2 ax cos" ax 



IGRALS INVOLVING tai 



14.429 I tanoarcZx = Incoaax = -Insecax 

J « a 

14.430 ftonZaajd* ^ t^EM _ ^ 

14.431 r tan* ax dx = ^^^ + ^ In cos ax 
J 2a a, 

14.432 rtan"a«sec2axdx - t&n^^^ax 
J (7i+l)a 

J ti 



14.433 I I^Si^dx = ^ In tan ax 
tan ax a 



1 4.434 I 17:7^::: = r In sin ox 



tan ax a 



M.435 J.tan^.. = ^ 1^ , <^ , ?^ , . . . , 



22"(22n-l)B„{ax)2" + i 
(271 + 1) ! 



+ 



14.436 J^dx = ax + Iggli + g<^ + ■■ ■ + ''"<'!" " ^^^.^^'^f""^ + 



9 ' 7B 



(27t - 1)(2tc) ! 



14.437 fxtanSaxdx = ^-^^2^^ + ^ In cos ax - -^ 



14.438 



/ 



dx 



px 



+ ., , — 57 In (.Q sin ax + p cos ax) 



p + 9 tan ax p2 + ^2 ^ ^(p^ + q^) 

14.439 ftan-axdx = tan"-iax _ C^j^n-ZaxOx 
J (n — l)a J 



INDEFINITE INTEGRALS 



81 



INTEGRALS INVOLVING cot ax 



14.440 r 

14.441 r 



cotaxdx ~ — In sin ax 
a 



cot^azdx = -i^tox _ ^ 
a 



14.442 
14.443 



J 

j cot" 

X 

14.445 r 



cot3 ax da = - S°^.BE - 1 in sin ax 



fix C8c2 oa; dx = — 



2a a 

cot" + ' ax 



{n + l)a 



14.444 I — dx = In cot a* 

cot ax a 



dx 1 , 

— 7 = In COS ax 

cot ox a 



^AAAJ^ C * J iJ (*«)^ (ax)" 
14.446 I X cot ax az = -^<ax — ^—~ Tor" 



14.447 



/ 



cot ax , 1 £Ei _ (ttx)^ 

X ax 3 135 



22«F„(ax)2''- 
(2n + l)! 

22«F„(ax)2"-> 



(2n-l)(2ji)! 

14.448 fxcot^oxdx = -?LS^^ + ^insinax -^ 
J o a* 2 

dx „ pg q 



14.449 



J 



p + g cot oar p2 + g2 o(p2 -|- g2j 

14.450 rcot«axdx = - cot"-' ax _ rcot"-2axdx 
J (n - l)a J 



In (p sin ax + q cos ax) 



INTEGRALS INVOLVING sec as; 



14.451 I secaxdx = - In (sec ax 4- tan ox) = — In tan (-—+-- 
J a ^ o V 2 4 

J 



14.452 



sec2 ax dx = 



tan ax 



14.453 fsecaoxdx = 5e£«fLtanax + 1 j^ (ggc ax + tan ax) 
J 2a 2a 

14.454 fsec-oo^tanaxdx = ^^Sl°± 
J na 

14 455 f_^5_ ^ sin ax 
J sec ax a 

r ,_. _ 1 J(ax)2 , (ax)* , 5(ox)8 

/ 



14.456 



sec ax dx = 



a2 I 2 "^ 8 "^144 (2n+2)(2Ti)! 



■iMJi^-w I sec ax J , ^ (ax)2 ^ 5(ax)^ ^ 61 (ox)" ^ , gt.(ag)'"' ^ 

14.457 I dx = inx + '-^ + -^ + -^^^ + ■•■ + ^^^^^ + 



X 



14.458 I X aec2 ax dx = - tan ax -\ s In cos ax 

J a a2 



82 



INDEFINITE INTEGRALS 



+ q COS ax 



14.459 r ^ = a_P f- 

J q + p sec ax q q J P 

14.460 (sec- ax dx = aeC-'^ax tanax ^ n-2 f ^^^^^ax dx 
J a{n — l) n — 1 J 



'*■» 



INTEGRALS INVOLVING cac ax 

14.461 I CSC axdx = - In (esc ax - cot ax) = - In tan ^ 

14.462 fcsc2aa:cte = -S^ax_ 
J a 

14.4«3 /csc=a... = _ esc .. cot „. ^ ^ ,„ ^„ ^ 

14.464 f CSC" ax cot ox dx = -Hc^L«£ 
J na 

14.465 f— ^5_ - _ COS ax 

J CSC ax a 

(ttx)3 , 7_(ox)5 , , 2{22"-i-l)B„(ax)2'> + i 




■'.! ••» 



1 4.466 J X esc ax dx = ^ ■{ ax + ^-7^ + 'i i ^L 4- • ■ • + 



18 1800 



(2n + 1) ! 



+ 



J 



14.467 I H^i^dx = _JL + ^+ 7(^_j^ ,,_ ^ ^ 



2{22n-i-l)F„(ax)2"-i 



ax 6 1080 

14.468 I X csc2 ax dx = — 

14.469 



(2n-l)(2n)r 



_ X cot ax , 1 , 

~ H 5 In am ax 



a a' 



J dx — * _ p r 
q -\- p CSC ax ^ 9 J 



dx 



P + g sin ax 



[See 14.360] 



14.470 fcsCoxdx = _cscn ^axcotax ^ n-2 fcsC-^axdx 
^ a(n — 1) n — 1 J 




INTEGRALS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS 



14.471 fsin-'-dx = x sin-> - + yJa^-x^ 
J a a ^ 

I X sin ' - 
J a 



14.472 



''^ - 'T-TJ"^" 'a + 4 



14.4/3 I x2 sin-i - dx = -r-sm ' — I ;; 

J a Z a V 

C sin-Mic/o) rf_ - « . (x/a)3 1 » 3(x/a)S 1 • 3 ■ 6(x/a)7 , 
J X a 2-3-3'''2-4-5-5^2'4-6*7*7 

f sin-'i 
J xa 

14.476 r^sin-i^j dx = x ( sin-i ^ j - 2a! + 2^52^7^ gi^-i £ 



14.474 



14.475 



Jx/a) J _ _sm 



in-i j x/a) _ 1 f a + y/a^-x^ 
a 



-f 



INDEFINITE INTEGRALS 



83 



14.477 
14.478 
14.479 
14.480 

14.481 

14.482 

14.483 

14.484 

14.485 

14.486 

14.487 

14.488 

14.489 

14.490 

14.491 

14.492 



/ 



J COS"'— da; = 
a 

z cos~^ - dx 
a 

\ x^cos-^-dx 
a/ a 

T cos"' (z/g) 

J X 

J cos~' {xia) 
x^ 



lecos"^- — yJa^ - x^ 
a 






- 1 £ _ x-\/a^ — x^ 
a 4 



3 a 9 



dx 

X 

dx 



fin. -J 
_ cos~' (x/a) 



sin~i (x/a) , 
dx 

X 



[See 14.474] 



dx 



( tan-i- 
J a 

\ a:tan-> - 



_ cos-'(g/g) + 1 In / « + A^a^ - x^ \ 

X a \ X J 

a:fcos-i-j - 2a: - 2y/a^~x^ eos"'- 



ztan-i- ^ ^In(x2+a2) 



da: = ^(x2 + 



o2) tan-i ^ - ^ 



^ X _ (z/a)3 (a:/a)5 _ (z/g)! 

a 32 "^ 52 72 + •■• 

a; a 2a \^ a;2 y 

xcot-i I + |ln(x2 + a2i 



rtan-Ma:/a)^^ 
./ X 

J tan-Mx/a) ^^ 

Jcot^'— dx = 
a 

/x cot~^ -dx = 
a 

Jx2 cot~i -dx = 
g 

Jcot-Mx/g)^ 



^) 



-i{x2 + a2}cot-i^ + ^ 

X3 



|ln^- J tan-Mx/a) ^ fS,e 14.486] 



_ _ cot-i(x/g) , 1 , fx^ + a^ 
~ X + 2^^" -^2- 



14.493 j sec-'- dx = 



X sec-i a In (x + \/«^ ~ «' 

X sec-' - + g In (x + V^^ ~ "' 



) < 



'x2-a2) 



-1 * ^ ^ 

sec ^ — < n 

o 2 

n < sec ' - < n- 
<i a 



14.494 i xsec-' * dx 
a/ a 



.-. X 




, X ^ IT 

sec-' — < 7; 
g 2 



< 

^ < sec-' — 
2 a 



< 



14.495 



/ 



x^ sec - ' — dx = 



•. 1 



x3 a: gxVx^ — a2 g^ , , , /-^ s. 

y ^^ a~ — 6 ~ y In (a: + V^^-a^) < 

^^ ■=««-! ^ _i. a* Vx2 — a2 , a^ /— 

T^'''^ a + 6 "^ yln{x + Va^2_„2) 



(in ' 



I > ' 



■]•. 



-1 ^ 

sec ' - 

a 



^'i 



n < sec ^ — 



84 



INDEFINITE INTEGRALS 



14.496 f^^^Af^da: = ^ In 
J X 2 



14.497 r^ec Hxia) ^^ 
J X- 



14.498 fcse-i-da; = 



14.499 f a; CSC- 1 -da: 



2'""^ + a: + 2.3.3 + 2.4.5.5 + 2T4T 

sec-"(3;/a) , \/a;2 — o^ „ ^ .a: 

< sec-i — 



1 • 3(a/x)5 1.3. 5(a/x)? 
2.4.5-5 "•" 2.4.6-7-7 



(a;/a) ^ \/a;2 - pg 
■ ax 



a<i 



sec"' (x/g) _ yf: 

X 



x^ ~ a2 



ax 



X CSC"' - + a In (a: + y/x^ — a^ 



X cac-^ a In (x + \/^ 

a:* _, a: , aV^^TT^ 
2 a 2 

a:^ , a: aylx"^ — a^ 
— CSC - ^ — i 

.3 a 2 



■a^] 



IT < sec-' - < IT 
2 a 

< csc-i- < ^ 

a 2 

-^ < cse-i - < 

1 a 



< csc-i - < ^ 
a 2 

n < CSC-' - < 
2 a 



14.500 r a;2 CSC-' -da; 
J a 



14.501 { esc Mx/g) ^^ 



-=- CSC ^ h — - — 

3 g 6 

x^ 



" ■ ^In(x + Va:2-g2) 



^ + -.n 



/ CSC 1 (g/g) 



K^ , X axyjx^ — a2 a,3 ^ 

y"^^"'a- 6 -|ln(x + V^^^) 

I a 



< 
< 



-1 * ^ iT 

CSC ' - < - 

a 2 



o < csc-1 - < 
£■ a 



<x 



^ (g/a;)3 1 . 3(g/a;)5 1 • 3 • 5(g/g)7 , \ 

2*3.3 "^ 2.4.5-5 2-4.6.7.7 *"/ 



14.502 I £^£lM^rf^ 



/ 



csc~' (a;/g) _ Va:^ — o- 
3^ ga: 



< 



14.503 


1 a:"" sin-* - dx 


14.504 


\ x"^ COS"* —da; 
J a 



_ g'»+^ 



a;m + l 



J 



14.505 I x^t&n-^-dx 



C5c-^(xla) yjx"^ — a^ 
X ax 

a:"' + i - _, ic 1 C -^ - , 

sm 1 7-rr I dx 

m + 1 a m+ IJ ^^2 - x^ 

— -— - cos ' — b r^ I — — dx 

m + 1 a m + ij^/^^--^ 

x^ + '^ . ^. X a C* K^ + i 
— - tan I — - I -5— — ^ da; 

m + 1 a m + lt'ai^ + a^ 



, a; . ;r 

CSC-* — < 77 

a 2 



^ < csc-J- < 
2 a 



14.506 f a;- cot-i ^dx = -^ cot"' ^ + -^ ( 4^ dx 
J a m+1 gm+ljx2 + g2 



14.507 I z^ sec-i^da: 



14.508 j a;"'csc-''-da: 



a:"' + ' sec-' (g/g) _ a C x*" dx 



./; 



m + 1 «t + 1 J y^^zr„2 

J z"' da; 
\/a;2 - a2 



a;m + i 3ec-i (a;/i 



sec-' (a;/o) _| a 

m+1 m + 



a;*""*"* CSC-* (xla) 



* CSC-* (x/a) , g_ 

m+1 m + 



- r^ 



'da; 



\/a2l^ 



a-m + i 



esc ' (x/g) 
m + 1 



g r a;'" dx 
m+1 J y^^IT^ 



< sec-'- 



0^2 



o < sec ' — 
d a 



< 



- < V 

a 



< 



CSC-' - < 5 
a 2 



K < cse-1 - < 
2 a 



INDEFINITE INTEGRALS 



85 



IGRALS INVOLVING 



14.509 



14.510 



J 
J 






xe'^dx = — (a:-- 
a \ a 



14.511 r»2««d^ = '—(^2^^+^ 



14.512 { x-efl^dx = ^:i^ ~ ^ C x^-ie<^ 
J a o. J 



dx 



= ^f ^n ^ ■»«""' , n(w-l)x"-2 



(-l)"rt! 
a" 



14.513 



14.514 



14.515 



14.516 



14.517 



14.518 



14.519 



14.520 



14.521 



14.522 



14.523 



14.524 



^ ^ 1-1! ^ 2-2! ^ 3-3! 

f — d* - -^°' + ^ f ^"^ ^, 



r rfa; X I 



J dx _ * I 1 1 



dx 



pC^ + qe"'^ 



— — tan->( -/£e<" 
aVP7 \ V 9 



In 



if n = positive integer 



. 2ay—pq \e^ + yl—qlv> 
{ ^?,mhxdx = ^'^i'^^iribx-bcoBbx) 

r e" cos bx dx = ^(a cos bx + b sin &a;) 
J a2 + 62 

r a:e<" sin bx dx = '^^'"(o sin bx - b cos ba) _ c<"{(o2 - 62) sin 6g - 2ab cos 6g} 

J a2 + 62 („2 _,_ 52)2 

f xe<^ cos bx dx = ^^'^^'^ '^"^ ^^^ + & s'" &a^) _ e^ICa" - 6^) cos 6x + 2ab sin 6a;} 
J a2 + 62 (a2 + 62)2 

r«-l„xd. = e'^lnx _1 fe^^^ 
J a a J X 

J e- sin" 6a. do: = £!^^in^(„ sinfcx - n6 cos6«) + ^^^^^ J e- sin-2 ft^ d^ 
j'6-cos«6a:d« = ^^ll^^^^^f^bx ^ nb ^r.hx) ^ "^^^^^ ^ e'^ co^^-ibx dx 



INDEFINITE INTEGRALS 



INTEGRALS INVOLVING In a: 



14.525 
14.526 
14.527 
14.528 
14.529 
14.530 
14.531 
14.532 
14.533 
14.534 
14.535 
14.536 

14.537 
14.538 
14.539 



dx = a; In X — :b 



Jin. 

J ^2 

x\nx dx — -z-{lnx — ^) 

Xaim+i / 1 \ 

X- Inxdx = ;j^^^p^ (^In X - -^j^j^ j [If m = -l see 14.528.] 

J X 2 

Jinx j^ _ _lil£ _ 1 
x2 X a; 

j ln2 a; dx = x In^ a; — 2a; In x + 2a; 

J in" X dx _ In" + ^ x 



n+ 1 



[If n = -l see 14.532.] 



f-T^ = In(lnx) 
J X In X ^ ' 

r^ = ln(lnx) + lnx + ill^ + il^+ ■■■ 
Jinx 2-2!3'3! 

Jinx 2*2! 3'3! 

I In^xdx = xln^ic — w | ln"~ia:!dx 

\ x^Xn^xdx = ^"'"^^^""'^ ^ rx'«ln"^ixdx 



If m = -1 see 14.531. 



j ln(x2 + o2)dx = xln(x2 + a2) - 2x + 2atan-i- 

rin(x2-o2)dx = « In (x2 - o2) - 2x + o In (|-^ 

rx"»ln(x2^a^)dx = x-^Mn(x^^a^)„ 2 C ^^^ 
J ' m + l wi+lJx2±o2 



INTEGRALS INVOLVING sinh aa; 



14.540 



J sin 



h ax dx = 



cosh ax 



14.541 fxsinhaxdx = xcoshox _ sinha£ 
J a a^ 

14.542 I x2 sinh ax dx = I 1 — g ) cosh ax — s ainh ax 



INDEFINITE INTEGRALS 



87 



14.543 
14.544 
14.545 

14.546 

14.547 
14.548 
14.549 
14.550 

14.551 
14.552 

14.553 

14.554 

14.555 

14.556 

14.557 
14.558 
14.559 
14.560 
14.561 



J X 3-3! 5-5! 



/sinHox , _ sinh ax , T cosh ax , ._ , , „^„, 

— ^2 — "* - ~ — "*" " J — X — f 14.565] 



J. 



x2 "- a: 

dx _ 1 , , , OiB 

-r — - In tanh -^r 

nnax a 2 



dx 



sinhax a^] 18 1800 



sinh^ ax dx 



2a 2 

_ X sinh 2ax cosh 2ax x^ 



dx _ coth ax 



J sin 

rsinh2oxdx = sinh ax cosh ax x 
J 2a 2 

Jxsi 

J sinh2ax ~ ^ 

r sinh ax sinh px dx = sinh (a + p)x _ sinh (a - p)x 
^ 2{a + p) 2(a - p) 

For a = ±p see 14.547. 

(' sinh ax sin px dx = « cosh ax sin px - p sinh ax cos px 
•^ a-2 + p2 

r sinh ax cos px dx = ° cosh ax cos px + p sinh ax sin px 
J a2 + p2 



2(-l)''(23"-l)gJax)2"+i 
"^ (2n + l)! "^ 






dx 



In 



qe 



_ *" + p - Vp2 + q2 

+ 9 sinh ax aV^TT^ "' V^e^^ + p + ^/^+^ 

dx — q cosh ax _i. P 



(p + g sinh ax)2 



+ 



h 



dx 



/ 



da: 



a(p2 + g2)(p + g sinh ax) p^ + q^ J p + q sinh ax 

^ tan-i Vq^ - p^ tanh ax 

ap\j(^ — p2 P 



p2 + g2 smh2 aa; 



p + Vp2 — gg tanh ax 
, 2apv'p^ — g2 \^p — \/p2 — qr2 tanh ax 



In 



/^ 



dx 



p + \/p2 + g2 tanh ax 



2 - g2 sinh2 ax 2apyf^^^^ \p - V^^TT^ tanh ax, 

J x*" sinh ax dx = x*" cosh ax _ m T j^_i cogj^a. j^j. [ggg 14.585] 

J„- i,„„, j„ _ sinh"^i ax coshax n — 1 f . ,,_„ , 
sinh" ax ax — I sinh'' 2 ^x dx 

TsinJ^^^ = -sinh ax ^ _g_ f £2^ ^^ [See 14.587] 

J X" (m — 1)3!"- > n — 1 i^ x" 1 L J 

/ dx — — cosh ttx _ w — 2 r dx 

sinh" ax a(n — 1) ainh"^^ ax n — 1 J sinh""^^^; 

/ xdx _ — X cosh ax 1 n — 2 r x d; 

sinh^ax a(n — 1) sinh"-' ax a2(m— l)(n — 2) sinh''~2ax n — ij ainh"- 



X dx 



8» 



INDEFINITE INTEGRALS 



INTEGRALS INVOLVING cosh aa; 



14.562 
14.563 
14.564 
14.565 
14.566 
14.567 
14.568 

14.569 
14.570 
14.571 
14.572 
14.573 
14.574 

14.575 
14.576 
14.577 
14.578 
14.579 
14.580 

14.581 

14.582 



i_ J _ sinh ax 
cosh ax ax = 

X cosh ax dx — 



a 
X sinh ax 



cosh ax 



x^ cosh ax dx — — 
cosh ax 



2x cosh ax 



+ ( — + ^ ) sinh ax 



X 

cosh ax 

x^ 



d, = in^ + i£^+4«^+i^ + 



2*2! 
dx = -SS^l^ + 



4-4! ' 6*6! 
sinh ax 



"J^ 



dx [See 14.543] 



cosh ax a 

xdx _ 1 J(ag) 



cosh ax 



(gg)^ , 5(ax)'> 

8 "^ 144 "^ 



, „ , X , sinh ax cosh ax 
cosh^ ax dx = -r- + ^ 



, „ , x2 a; sinh2ax 

X cosh^ ax dx = -r + : 

4 4a 



cosh 2ag 
8a2 



dx 



tanh ax 



cosh^ ox 



J 

J 

X 

/ 
J 
/ 
J 
J 
J 
/ 
J 

/, a sinh ax sin px — p co sh ax cospx 
cosh ax sin px dx = 2 , n 

I cosh ax cos px dx = 

/ 
J 
/ 
J 
J 

J (cosh ax — 1)'^ 

/ dx . 

p + 17 cosh ax 

/ dx 
{p + Q cosh ax)2 



(-l)"B„(ax)2« + 2 
"^ (2n + 2){27i)! ''■ 



t. J _ sinh a-p)x . 3mh(o+p)a! 

cosh ax cosh px dx = — -rr-^ ^^ 1 n- , „■■ — 

2(a — p) 2(a + p) 



_ a sinh ax cos px + p cosh ox sin px 

a2 + p2 



dx 



cosh ax + 1 


dx 


cosh ax — 1 


xdx 


cosh ax + 1 


X dx 


cosh ax — 1 


dx 


(cosh ax + 1)^ 


dx 



1 . , ax 
= — tanh-Tp 
a 2 

1 ., ax 

= coth -^ 

a 2 

X , ax 2 , , ax 

= - tanh ^ - -5 In cosh -5- 



a 2 a^ 6 

= 2^*^"*^T~6^'^"*' 2 
^ J-coth^-icoth3^ 



2a 



6a 



2 ^ _, Qe'^ + P 



In 



ge' 



or + p — '\Jj^ — q^ 



q sinh ax P r_ 

a(q2 - p2)(p + g cosh ax) q^ _ p2 J p 



dx 



+ g cosh ax 



INDEFINITE INTEGRALS 



89 



14.583 



J P'- 



dx 



14.584 

14.585 
14.586 
14.587 
14.588 
14.589 



h 



(p- cosh^ ax 



dx 



In 



/ p tanh gg + Vp^ - q^ \ 
Vp tanh ax — Vp^ ~ 9^/ 



2 + ^2 cosh^ ax 



2apy/f^ — q2 \^p tanh ax — Vp^ ~ 9^ 
—1 . p tanh ax 

- rn-n^J 

1 



V^23^ 

p tanh ax + Vp^ + g^ 
2apVp^ + ^ \p tanh ax - y/p^ + q^. 



In 



1 



Vp2 + 



tan-1 



j x"" cosh ax dx = 

J 



ap 



x"" sinh ax 



p tanh ox 
\/p2 + g2 



_ m /* 



cosh- ax dx = cosh"-'axsinhax ^ 



"-^J 



x"-' ainhox dx [See 14.557] 

cosh" ^2 da; dx 
Tcoshox^ ^ -cosh ax _a_ Tsinhox ^ [See 14.559] 

f dx _ sinh ax n — 2 T da 

cosh" ax a(n— 1) cosh""' ax m— 1 J cosh" 



dx 



/ 



cosh" ax 
X dx 



:.flf r. 



^',. t. 



X sinh ax 



+ 



cosh" ax a{n — 1) cosh"" ' ox (n — l)(n — 2)a2 cosh'»~2 ^x 



n-2 r xdx 
n — 1 J c03h"~2 ax 



INTEGRALS INVOLVING sinh ax AND cosh aa; 



14.590 
14.591 
14.592 
14.593 
14.594 
14.595 
14.596 
14.597 
14.598 
14.599 
14.600 
14.601 



sinh ax cosh ax dx = 



sinh^ ax 
2a 



J. . V J _ cosh (p + tf)x , cosh (p — g)x 

ainh px cosh qx dx = — -/^^ ^ + — 57^^= — p- 
2(p + g) 2(p - g) 

C sinh" ox cosh ox dx = 8mh" + 'ax ,jj ^ ^ _j see 14.615.] 



. . n LD J _ sinh 4ax x 

sinh^ ox cosh^ ax dx = — -rr — -z 

32o o 



dx 



= - In tanh ax 



sinh ax cosh ax 

^^ = -^Uin-^Binhax-S^h^ 

sinh^ ax cosh ox a a 



r cosh" ax sinh ox dx = cosh" + Ux [if n = -1, see 14.604.] 

J (n + l)o ^ 

J 

J sii 

J 

/ CO 
si 

J 



dx 



sinh ax cosh^ ax 
dx 



sinh^ ax cosh^ ax 



55^*1^ + ilntanh^ 
o a / 

_ 2 coth 2ax 
a 



sinh^ax , sinh ax 1 . _, ._. ._ 

— : dx = tan ' smh ax 

cosh ax a a 

=.25h!°£dx = ^2^^^^ + I In tanh ^ 
sinh ax o a ,2 



dx 



cosh ax (1 + sinh ax) 



1 j„ /I + sinh ax \ + 1 ^^^_, ^^ 
Za V cosh ax I a 



I'.' ■ 



J. ' 



90 



INDEFINITE INTEGRALS 



14.602 
14.603 



J si 



dx __ J^ ]j^ tanYi — + ^ 

sinh ax (cosh ax + 1) 2a 2 2a(cosh aa; + 1) 



dx 



14.604 
14.605 
14.606 
14.607 
14.608 
14.609 
14.610 
14.611 

14.612 
14.613 
14.614 



J 



1_ . ^^ _L a£. _ 1 

sinh ax (cosh ax — 1) 2a 2 2a(cosh aa: — 1) 



INTEGRALS INVOLVING tanh ax 



lanh ax dx = — In cosh ax 
a 



tanh2 ax dx = x — 



tanh aa; 
a 



( t^nh^axdx = ilncoshax - ^^"^'^^ 



2a 



f tanh" ax sech2 oa; da: = *^"^''V.°'' 
./ (n 4- l)a 

X 

J 



sech^ ax J _ 1 ■ . . 

; aa: = - In tanh aa; 

tanh ax a 



—. = - In sinh ax 

tanh ax a 



J 

r tanh 

J 



* V . 1 J(ax)^ ((ia:)= , 2(ax)7 

tanhaxdx = ^^1^-^+^ 



2 3 



xtanh^axdx = ^ - ?-l5n»L«^ + X in cosh ax 



(-l)n-122«(22n - l)g„(gx)2n + l 
(27t+l)! 



+ 



ax , (ax)3 , 2(ax)'' 

-dx = ax--y-+^5- 



dx 



px 



p + qf tanh ox P^ — 9^ a(p2 — q^) 



(-l)"->22'"(22n-l)B„(ax)2''-i 
(2n-l)(2Ti)! 



In (q sinh ax + p cosh ax) 



+ 



14.615 
14.616 
14.617 
14.618 
14.619 
14.620 



r tanh" ax dx = tanh" ^ ax ^ T ^^^y^n-z ax dx 
J a(n-l) J 

^^^^^^ ^NTlolBn^NVOmN^otra 

/ 
J 



coth ax dx = - In sinh ax 
a 



coth^ ax dx = x — 



coth ax 



Jcoth»axdx = ilnsinhox - ^°*^^''''= 



2a 



rcoth«axc8ch8oxdx = -S^^^lll^ 
J (n + l)a 



csch^ ax . _ 1 , .. 

— dx = In coth ax 

coth ox a 



— ^ — = - In cosh ax 
coth ax a 



INDEFINITE INTEGRALS 



91 



14.621 I X coth ax dx 

14.622 C X coth2aa;dx 



\\ax + ^^-^^^ 



(_l)n-122nBJax)2" + l 



225 



(2n + 1) ! 



+ 



_ X X coth ax 



+ — 5 In sinh ax 



14.623 



14.624 



{2n-l)(2TO)! 



C coth ax ^^ ^ _ ± + ^ _ (gg)^ .... (-l)"22"B„(ga;)2''-i 
J X ax 3 135 "* (2n -lV2-n.il 



+ 



+ g coth aa; 
14.625 rcoth''a«dx = 



p2-qz - a(p2 - ^a) ^" (^ »'"*» az + g cosh ax) 
coth"~* ax 



ain — 1} 



J 



+ I coth" •'^ ax dx 




^^^^^^^^^^^^tAtS mvOLVfNG sechaa: 

1 4.626 I sech ax dx = — tan~ ' e°^ 
J a 

14.627 fsechaaxdx = ^"^°' 
J a 

14.628 Tsech^axdx = sechax tanhax ^ 1 ^_ 
./ 2a 2a 

14.629 fsech^oxtanhazda; = _ sech^L"^ 
•^ na 



14.630 f —S^^— - sinh gg 
J sech az a 

14.631 fxsechaxdx = A W _ W^. , S(ax)6 (-l)«g„(axpn.2 
^ a2 [ 2 8 ^ 144 ^ (27i + 2)(2m)! 



+ 



14.632 r X sech2 axdx = ^^"^"^ - ^ in cosh ax 

14.633 fsechoa ^^ ^ ^^^ _ (^)2 Siax)-* _ 61(ax)« (-l)"g^(ax)2" 

J SB 4 96 4320 "^ 2n(2K)! "'" 

14.634 r ^ ^ ^_P f dx fS*.pi4KRii 

J 9 + P sech ax g Q J p + q cosh ax ^ ^ ^^'^^^^ 

14.635 fsech" axdx = sechn-2 ax tanh ax ^ ^^ r ^^^^„,, ^^ 
•-' a(n — li ■» — II 




INTEGRALS INVOLVING 



14.636 



14.637 






csch ax dx = - In tanh — 
a 2 




csch^ ax dx = — 



coth ax 



14.638 r cschs azdx =^ - csch ax coth ax _ ^ i ^^ , ax 
J 2a 2a 2 

1 4.639 f csch" ox coth ax dx = - ^s^^" '^^ 
f' na 



INDEFINITE INTEGRALS 



14.640 



J 



'^^ = -cosh arc 



csch ax 



,4.*4, ..csch»x^ = -l„._<^+?^ + 



14.642 
14.643 
14.644 
14.645 



X csch2 axdx = ~ £cothax ^ 1 j^ ^j^j^ ^^ 



/ 
/ 

J csch Q 

r 

J g + P 



_1^n/')2n-l_ 



+ 



2(-l)''(2 



l)B„(aa;)2"+i 



(2m + 1) ! 



+ 



oar 6 "^ 1080 



= 5. _ p r 

*7 9 J 



da; 



csch ax Q 9 -' p + g sinh aa; 

— csch" ~ 2 ax coth aa; 



(-l)"2{22'«-i - l)fi„(ax)2«-i 
(2«-l){2n)! 

[See 14.553] 



+ 



csch" ax dx = 



a(n-l) 



_ « -2 r 

n - 1 J 



csch" ~ 2 ax dx 



InTEoBS? INVoVviN^mVERSEHYPERBOUcflulmlOl 



14.646 
14.647 
14.648 

14.649 



/sinh~^ — 
a 



da: = 



X sinh~' vxM-a2 

a 



X sinh~* — dx 
a 



f x2 sinh-» - d3B 



t+ti-''-*t- 



a: scVk^ + a* 



= — sinh"' — h 



X ^ (2a2 - a:2) Vx^T^ 



J 



sinh~' (a;/o) j^ 
* 



? _ <g/'>)^ , 1 ' 3(x/a)5 _ 1 ' 3 ■ 5{x/ay , ... 
2'3'3 2*4'5'5 2*4'6'7*7 



a 

ln2 {2z/a) 
2 



(g/x)^ , 1 ' 3(a/x)< 
2-2-2 2*4'4-4 



1 ■ 3 ■ 5(g/g)» 
2*4-6-6'6 



+ 



ln2 (-2g/a) (a/x)a _ 1 ■ 3(g/a!)< 1 ■ 3 ' 5(a/x)« _ 



2-2-2 2'4'4'4 "*" 2 -4 -6 '"6 '6 



14.650 fsinh-Mx/g)^^ ^ _ 



14.651 

14.652 

14.653 
14.654 

14.655 
14.656 
14.657 
14.658 



J 

; 

X 

J 



sinh~' (z/a) _ 1 1_ 



a+ \Ax2 + o2 



cosh 1 - da; = 
a 



X cosh"^ — dx 
a 



flc2 cosh""' -dx 
a 



X a V X 

xcoBh-'(x/o) - Va;2 - o2 , cosh-Ma:/a) > 



r co3h-'(x/a) ^^ 
+ if cosh-' (x/a) > 



[ X cosh-' (a:/a) + -^x^ - a^ , cosh"' (x/a) < 

ri(2x2-a2) cosh-' (x/a) - ^xVar^ - o2 , coah-'(x/a) > 
[i(2x2-a2) cosh-' (x/a) + ^xy/x^ - a^ , cosh"' (x/a) < 
r^x3 cosh-' (x/a) - ^(x2 + 2a2) yfx^ - aZ . cosh- Hx/a) > 
1^x3 cosh-' (x/a) + ^(x8 + 2a2) ^x^ - a^ . cosh"' (x/a) < 

= ^[|ln2(2x/a) + i^^+4:4i^ + UA4<^ + 



2.2-2 
0, - if cosh-' (x/a) < 



2.4-4-4 2-4.6-6-6 



\x\<a 
x> a 
X < —a 



cosh~' (x/a) ^ _ _ 



tanh-' -dx = 
a 



r cosh-' 

J 
J 
J 



_ cosh-' (x/a) _ 1, / a + Vx2+aA h if cosh-' (x/a) > 0, 
" '" ' ' + if cosh-' (x/a) < 0] 



X tanh-i - dx 

a 



xtonh-'- + ^In(a2-x2) 



^ + i(x2-o2)tanh-»f 



x^ tanh-' — dx 
a 



= f + ?'-''-| + f-(-"^-^") 



INDEFINITE INTEGRALS 



14.659 



14.660 



14.661 



14.662 



tanh-' (x/a) 



dx = 



— ^ 
a 



+ f^/°)^ , (x/a)^ 



tanh-i jx/a) 



J 
/ 

J coth-i ^dx = z coth-1 X + I In (ar2 - a2) 



X ^ 2a " \ a2 - a;2 



a:^ 



coth-i - dx = 



14.663 f x2 coth-i ^ da; 

14.664 poth-Mar/g) ^^ 

14.665 f coth-i (ar/g) ^^ 
J x^ 



'' a 

_ ax^ , x^ .,, X , a^ 






32 



52 



+ 



14.666 



14.667 



14.668 



14.669 



/ 



sech-i -dx = 



/^ 



sech-i^dx 
a 



f 



sech-i (x/a) 



dx = 



csch-i -dx = 



a 

.-1 X 



1 4.670 ( X csch - 1 - rfa; = 
J a 



14.671 



r csch~^ (g, 



/g) 



dx 



14.672 C xrn ainh-^ ^ dx 
J a 



_ cQth- '(ar/tt) , 1 , 

a; sech-i (a;/g) + o sin-' («/«), sech->(3;/a) > 
a: sech^Mx/g) - a sm~^ [x/a), sech->(ar/g) < 

J^9;2sech-Ma;/a) - laVa^ - x^, sech-Hx/a) > 
[ ^x2 sech-i (z/o) + ^gVa2 - ^2, sech-Ma;/a) < 

^ 2.2-2 2.4-4M 

i In {a/x) In (4a/x) + i^^^ + ^ ' ^(x/g)^ , 

I." 2*2-2 ^ 2M'4-4 

« cach-i ^ ± a sinh-i | [+ if a: > 0, - if x < OJ 

T "^^"^^ ^ - ~^ [+ if X > 0. - if X < 0] 

' ^ In (x/g) In (4g/x) + li^l^ _ 1 ' 3(x/a)^ ^ _ _ 
2'2'2 2»4'4'4 

^ 2-2.2 ^ 2-4-4-4 

_g ^ (g/x)3 _ 1 ■ 3(a/g)5 

a; 2*3'3 2'4"5'5 " 

- »"''*"^ _. a; J__ r a;m + i 



seeh-i (x/a) > 
sech-i(a;/a) < 



< X < a 
-g < X < 

1x1 > a 



sinh~i 

g m 



14.673 j x"* cosh-* - dx = 

14.674 I I-" tanh-' ^ dx = 
•^ g 

.4.675 J 



m + 1 

— - cosh-i 

m + 1 a 



^I^V^^^^ 



m 



hi 



a2 



X>n + l 



m 



x"* coth-i — dx 
a 



14.676 r ajwi sech-' - dx = 
-^ » ^:;^sech-^-^r^ 

14.677 C x'"csch-i^dx = -Ell^csch-i^ ± — ^^ r_£ 
•^ a m+1 am+ljr- 



+ 1 ^ a^ ~ x^ 
dx 



cosh-i (x/a) > 
cosh*-i (x/a) < 



m + 1 
m+1 a m + 1 J ^^ 



X 

+ 
7n 



Vo^ - x^ 



'dx 



\/g2 - x2 

'dx 



\/x2 + a2 



sech - 1 (x/a) > 
sech-'(x/o) < 

[+ if X > 0, - if X < 0] 




DEFINITE INTEGRALS 




DEFINITION OF A DEFINITE INTEGRAL 



Let f(x) be defined in an interval a ^ x S b. Divide the interval into n equal parts of length Aa; = 
(b — a)/n. Then the definite integral of /(*) between x — a and x = b ia defined as 

15.1 f fix)dx = lim {f{a)Ax + f{a + ^x)^x + /(a 4- 2Aa;) Ax + ■•■ + /(a + (n - 1) Aa;) Aaj} 

The limit will certainly exist if f{x) is piecewise continuous. 

If fix) = -raix), then by the fundamental theorem of the integral calculus the above definite integral 

ax 

can be evaluated by using the result 

15.2 J f{x)dx = j i^s{.x)dx = 



g{x) 



= 9{h) - g{a) 



If the interval is infinite or if fix) has a singularity at some point in the interval, the definite integral 
is called an improper integral and can be defined by using appropriate limiting procedures. For example, 

15.3 C f{x)dx = lim f fix)dx 

15.4 f fix)dx = lim r f{x)dx 

Xb y*b-e 

fix) dx = lim 1 fix) dx if 6 is a singular point 

15.6 I fix) dx = lim I fix) dx if a is a singular point 

OENf^UFBRMfftA^NVofvTN^WBmTinwTfl 



15.7 
15.8 
15.9 



^b pi} fh fb 

J (fix) ± gix) ± k{x) ± • • •> da: = J fix) dx ± J gix) da; ± J h{x) dx ± ■ ■ • 

I c fix) da; = c j f{x) dx where c is any constant 



f /(a;)da; = 

15.10 r /(a;)da: = - C fix)dx 

15.n f fix)dx = rf{x)dx+ C fix)dx 

a a c 

f(x) dx = (b — a) fie) where c is between a and 6 

This is called the meayi value theorem for definite integrals and is valid if f{x) is continuous in 
a^ x^b. 



94 



DEFINITE INTEGRALS 



96 



15.13 



J /(x) g(x) dx ^ /(c) J g{x) dx where c is between a and 6 

This is a generalization of 15.12 and is valid if f{x) and g{x) are continuous in a -^ x ^ h 
and p(a;) ^ 0. 



15.14 



lEIBNITZ'S RUIE FOR DIFFERENTIATION OF INTEGRALS 

*'*,ttt> »'*i(a) "" «« aa 



^PSOXlWATrft^RMlilAlSoR DEFINITE^ »^^^ 

In the following the interval from x = a to x = b is subdivided into n equal parts by the points a = x^, 
«!. X2, . . ., a:„_i, a:„ = 6 and we let y^ =^ fix^), y^ = /(a;,), y^ = /(arg), . . ., y„ = /(ic„), /t = (6 - a)/n. 

Rectangular formula 



15. 



r'' 

IS J /(x) da: « A(yo + yi + y2+ •■■ + y„-i) 



Trapezoidal formula 



15.16 



j; 



fix) dx 



^iVsi + 2j/, + 2i/2 + • • • + 2|/„_i + ?/„) 



Simpson's formula (or parabolic formula) for n even 



15.17 



J f{x)dx « -g-d/o + 43/1 + 23/2 + 4^3 +-■■ +2i/„_2 + 4y„_i + !/„) 



IN6 RATIONAL OR IRRATIONAL EXPRCSSIOI 



15.18 



15.19 



15.20 



15.21 



15.22 



15.23 



15.24 



15.25 



X" 



dx 



x2 + a2 



2a 



X 



sinpiT- 



J„ 1 + 



J'" x'^dx _ 
„ a;" + a" " n si 



< p < 1 



sin [(m + IV/n] ' 



< m+1 < n 



x"^dx 



jr sinmff 



+ 2ar cos (i + x^ sin m-rr sin fi 



dx 



»^o y/cfi - x^ 2 

Va2 - a;2 d-c - 



I 3;"'(a" - a:")Pda: = j ^*' / 



I (^" 



x'^dx 



vV[{m+\)/n-\- p + 1] 

(-l)r-I^ttm + l-nrr[(m + l)/n] 



+ a")'- nsin[(m + l)Wn](r-l)!r[(7n+l)/m - r + 1] 



< mi + 1 < Mr 



96 



DEFINITE INTEGRALS 



DEFINITE INTEGRALS INVOLVING TRIGONOMETR 

All letters are considered positive unless otherwise indicated. 

m,n integers and m ^^ n 



m 



S ''" 



15.26 I sin Tnx sinnx dx = 

1 jr/2 m, n integers and m = n 



mi, M integers and m ¥= n 
15.27 I cos mx cos nx dx == -s 

7r/2 m, n integers and m = n 



f 

15.28 I ainmxcoanxdx — 



m,n integers and m + n odd 

2m/(m'^ ~ n^) m, n integers and m + n even 



1 5.29 I sin2 xdx = i cos^ x dx ^ ^ 







sinS"" x dx 



15.31 r sm2^+^xdx = P cosZ'^+ixdx = ^„'^_'^": ^^ . , m = l,2,... 



''o 

sin^P-ixcos^-i-ixdx = J^P^ ^<'^^ 
2r(p+(/) 

15.33 r ?mp^d« = ^ 



cos2"'a:da: = ^ '■^ '.^ '." ^'!!~ ^ ^ . m = 1, 2, . . . 



2 • 4 • 6 ■ • ■ 2mi 2 



1 • 3 • 5 ■ ■ ■ 2m + 1 



15.32 



B-/2 p > 
p = 
-5r/2 p < 



15.34 



/: 



sm px cos qx 

x 



dx — 



I 



15.35 f sin pa: sin ga;^^ ^ 



a^J 



p > g > 

:r/2 < p < qf 
jr/4 p = g > 

7rp/2 < p ^ g 
7rg/2 p ^ Q > 



15 



sin^ px ^^ _ £p 



- cos px .^ _ ffp 



x^ 



da; = ^ 



.36 r° 
15.37 ri 

15.38 r"cospx-cosqa; ^ = i„ £ 

J, « p 

15.39 r" cos px - cos qx ^^ _ Trig - p) 

V 

15.40 CSB^^dx = ^e-- 
J„ x2 + „2 2a 



15.41 



X°° X sinm 
x2 + a: 



mx ,„ _ 7r„_ 



dx - -^e-"*" 



X x(x2 + a2) 



- ^d-^-""") 



15.43 



15.44 



15.45 






dx 



+ 6 sin X 



n a 



dx 



'o a + 6 cos X 

.ff/2 



n a 



dx 



2n- 




Va2- 


62 


2ff 




Vo2- 


62 


COS" 


(6/o) 





+ 6 cos X yo2 — 62 



DEFINITE INTEGRALS 97 



15.46 r ' ^^ = C Ja; ^ 2r-a 
J^ (a + 6 sin x)^ J^ {a + b cos a;)2 (a^ - 62)3/2 

15.47 T" ^ ^ 2:r 0<a<l 

15.48 f g sin a; da; ^ \ (v/a) \n (1 + a) \a\ < 1 
J„ l-2acosx + a^ | . In (1 + 1/a) |a| > 1 

,5.49 r cos.»xd. ^ _^a™^ ^,^ m- 0.1,2.. 

^Q 1 — 2a cos a; + a2 1 — a- - , ■ 

15.50 J sinoK^da; - J cos 0^2 da; = \'\[¥' 

15.51 I sin aa:" da: = _1^ r(l/M) sin -^ , n > 1 

15.52 I cosaa:" dx = — Vr l '1/") cos — , n > 1 
J^ »mi/" 2t[ 

15.53 r^inx^^ ^ fsSlSdx = .fl 

15.54 r^inx^ ^ „ , ^ :" _, 0<p<l 

J(, a:P 2r{p) sin {pn-/2) '^ 

15.55 r^^^dar = ——J!——-, 0<p<l 
Jg x» 2r(p) cos (pW2) ^ 

1 5.56 I sin ax^ cos 26a; da; — — ■\l-^ ( cos sin — 

J(, 2 \ 2a \^ o a 

15.57 I cosaa!^ cos26a: dx = 7: A /tt" I cos (- sin — 

J^ 2 Y 2ay a a 

"■58 J^ _^d. = - 

15.59 r^il^dx = ^ 
Jo ^' 3 

15.60 r^Hl^rf^ = i 
J,. X 2 



15.61 





. 1" 



dx 



.Q . + tan"'x 4 



,5.63 /;^H^.. = J,-i+l-i+.. 



^dx - I In 2 



15.64 C ^iill^ 

15.65 p l-cosx ^^ _ rcosx^^ ^ 

Jo a; J^ X 

1 5.66 I ( T — I — ■= — cos X 1 — = y 

15.67 f^tan-ipx-tan-igx ^^ = £ ]„ £ 
J„ X 29 



98 



DEFINITE INTEGRALS 




DEFINITE INTEGRALS INVOLVING EXPONENTIAL FUNCTIONS 




15.68 
15.69 

15.70 
15.71 
15.72 
15.73 
15.74 

15.75 
15.76 
15.77 
15.78 
15.79 
15.80 

15.81 
15.82 

15.83 

15.84 

15.85 
15.86 



Jg-ar cOs6x dz = 


e-" sin bx dx = 

X 
X 

e-"^' COS 6a: da; = i../?e-b« 
2 \ a 



a2+ (,2 

& 

a2 + 62 



- e-°-sin6x ^^ = tan-i^ 
X a 

dx = In - 



-b'/4a 



r"e-""' + f^ + -'da: = i - f^efb^-iacJ/to ^^ _6_ 
where erfc (p) = — | e-^ dx 

C x'^e-'^dx = ^<^+l> 

X' 



*-4ac}/4a 



_^ , _ r[(7n + i)/2] 



x^e""* dx = 



l'^-'"'*"^'^ =\y[l 



gatm + i'/z 



/■ 



xdx _ X-L-L-l-L + JLj- 

e' - 1 12 "^ 22 "^ 32 "^ 42 "^ 



6 



For even n this can be summed in terms of Bernoulli numbers [see pages 108-109 and 114-115]. 

r 



xdx ^ 1 I . J^ _ A . 

e^ + 1 12 22 32 42 



12 



For some positive integer values of n the series can be summed [see pages 108-109 and 114-115]. 

r 



sinmx ^^ ^ 1 oth^- J- 



g27rl _ X 



2 2»i 






DEFINITE INTEGRALS 



15.87 



15.88 



/: 



r^ 



X secpx 2 \a^ + p^ 



dx = tan~i tan~i - 

CSC px p V 



15.89 r"^-°^a-co5a^)rf^ ^ cot-la- fin (a^+l) 



ITE INTEGRALS INVOLVING LOGARITHMIC FUNCTIOI 



15.90 { x^(\nx)^dx = , ^ ,^^"'!' w> -1, n^0,l,2,... 



If 717^0,1,2,... replace m! by r(n+l). 
15.91 17 



da: - ^2 



+ X 



15.92 



x 6 



J, 1-: 

15.93 C'JLiLt^dx = 4 
J, 0. 12 

15.94 I'lnO^ 



dx = -- 



15.95 C lnx\n(l + x)dx = 2-21n2-^ 

15.96 r lnxln(l- 
.-'ft 



x)dz = 2 - J 



15.97 



15.98 



15.99 



pxP_nnx^^ = -irZcBCpjrCOtpTT 0<p<l 



— dx — In —r 

In X n + 1 



I e-'\nx 



dx = —y 



V^ 



15.100 r e-^Mnxdx = -■^{7 + 2 1n2) 



dx = -T- 
4 



dx = -5 In 2 



15.102 I In sina; dx = I In cosx 

(ln8inx)2dx = I (lneosx)2dir = |(ln2)2+|^ 
15.104 r xlnsinxdx = "T^^^ 



15.105 



XTT/2 
sin a; In si 



sin X dx ~ In 2 — 1 



J»27T /^2Tr 

In (a + 6 sin x) dx = I In (a + 6 cos x) dx = 2o- In (a + V^^^) 
(1 •'n 



100 



DEFINITE INTEGRALS 



15.107 yin(a+bco^x)dx = ^ln(^£±v5!E^ 



15.108 j In (o2 - 2a6 C9S a: + 62) da: = 



2n- In a, o ^ 6 > 
2n- In 6, 6 ^ a > 



15.109 



15.110 



15.111 



I In (1 + tan x) dx - ^ In 2 
Jf, 8 

r sec a; In ( f^-^-^^^ ) rfa: ^ A{(cos-i a)2 - (cos-^ 6)8} 






2.i„|,,. = _,i||i+?i||£H.EnSa^ 



See also 15.102. 




DEFINITE INTEGRALS INVOLVING HYPERBOLIC FUNCTIONS 



15.112 






4^1;^ dx = ;^tanh^ 
Sinn ox 26 26 



15.113 



X 



15.114 f 



COS ax , _ TT . oir 
— ; — ; — «* = :rr seen rrr- 
cosh hx 26 26 



sinh aa: 4a2 



15.115 



fii 



a;" da; _ 2"+i - 1 . . .,. | 1 . 1 1 

sinhaa; ~ 2"a«+i ^ I" "^ -^M x-i + i "^ 2"+' "^ 3» + » "* 



If M is an odd positive integer, the series can be summed [see page 108]. 

15.116 r^^dx = ^C8c^-f 
I e^x -1-1 26 6 2a 



15.117 r"^h«^da: = f-^cot¥ 
J ebz _ 1 2a 26 6 




ISCELLANEOU5 DEFINITE INTEGRAL 



M 




,5., ,8 J^/W^AMrf^ ^ (/(O) -/{-)} In ^ 

This is called Frullani's integral. It holds if fix) is continuous and I '^^' ~ '^'^' dx converges. 



15.120 r (a + a;)'"-i{a-a:)''-ida; = (2a)'" + n-ilMi:M 
^~„ rim + n) 




16.1 



16.2 
16.3 



DEFINITION OF THE GAMMA FUNCTION T{n) FOR n>0 



r( 



n) = ( f-^e- 

^ n 



*dt « > 




r(n + l) = nr{n) 

r{n + l) = nl if n - 0,1,2, ... where 0! - 1 



THE GAMMA FUNCTION FOR n<0 



For n < the gamma function can be defined by using 16.2, i.e. 



16.4 



n 



GRAPH OF THE GAMMA FUNCTION 



r(M) 


_ n .' 1 


U-^ t 


^^t -/ 




— 1 -S -1 -1 1 > S 4 I 

„ -I 


it ^ _. 





Fig. 16-1 



16.5 
16.6 
16.7 



SPECIAL VALUES FOR THE GAMMA FUNCTION 



rim+i) = l'3>5.^-^(2n.-l) ^ ^. 1.2.3, 



(_l)m2m^ 



r(-» + i) = ^^3^^ 



(2m - 1) 



m- 1,2,3, 



101 



102 



THE GAMMA FUNCTION 



RELATIONSHIPS AMONG GAMMA FUNCTION! 



16.8 
16.9 



r(p)r(i-p) = 



sin pn- 



22*-* r{x) r{x + ^) = V^r(2x) 
This is called the duplication formula. 



16.10 



For m = 2 this reduces to 16.9. 




OtHErDteFrMlttONS OF THE GAMMA PUNCtrON 




16.11 



16.12 



^^^+^^ = ji"L (.+iK;V2V---t+fe) 



fe* 



This is an infinite product representation for the ^amma function where y is Euler's constant. 



DERIVATIVES OF THE GAMMA FUNCTION 



16.13 



:"(!) = j e-* In K da: = -y 



16.14 



r(x) ^ ll =»/ \2 *+ 1 



+ ■■- + ^- 



n X + n~ 1 



+ 



16.15 



^NS FOR THE Gi 



r(« + l) = ■\/2vXX='e-^ ^1 + r^ + ^ 



139 



12x 288a;2 51,840x3 



+ 



This is called Stirling's asymptotic series. 



If we let X = n a positive integer in 16.15, then a useful approximation for n\ where n is large 
[e.g. 71 > 10] is given by Stirling's formula 

16.16 n\ ~ \/2^n«e-'' 

where — is used to indicate that the ratio of the terms on each side approaches 1 as n-* «>. 



16.17 



|r(tx)|2 = 



X sinh vx 



DEFINITION OF THE BETA FUNCTION B{m,n) 



17.1 



B(m,n) = J (^-^l-t)"-i dt m > 0, n > 



RELATIONSHIP OF BETA FUNCTION TO GAMMA FUNCTION 



17.2 B{'m,n) = iMliZii 

r(m + n) 

Extensions of B(m, n) to m < 0, n < is provided by using 16.4, page 101. 




ME IMPORTANT RESULTS! 



17.3 



B{m, n) = B{n, m) 



17.4 



B(m 



,n) = 2 I 



ir/2 



gJn2n"-l « cos^"-! 9 rffl 



17.5 



B(m 



'"^ - X (l+V-^« 



d« 



17.6 






103 



18 



BASIC DIFFERENTIAL EQUATIONS 
and SOLUTIONS 



...^Mmmi^^dLMOisAnou 


SOIUT.ON : , :!Hi^ 


18.1 Separation of variables 


1 , , .dx + 1 —r^dy - c 


fi(x)giiy)dx + f%(x)g2{y)dy = 


18.2 Linear first order equation 


ye/Pdx = j Qe Jpditia; + c 


1^ + P(x)y - Q(x) 


18.3 Bernoulli's equation ■- ;' 


where i" = y^""". If n = \, the solution is 
In y = \ {Q-P)dx + c 


g + P(x)y = Q{x)y^ 


18.4 Exact equation 


\ Mdx + (In - J- ( Mdx\dy = c 

where Bx indicates that the integration is to be performed 
with respect to x keeping y constant. 


M{x, y) dx + N{x, y)dy = 
where dM/dy = dN/dx. 


18.5 Homogeneous equation 


, C dv ^ 
J F{v) - V 

where v = y/x. If F{v) = v, the solution is y =^ ex. 


f = <f) 



104 



BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS 



105 



OlFFiRENTIAL EQUATION 



18.6 



y F{xy) dx + X G{xy) dy — 



!• ^ Linear, homogeneous 
second order equation 



d^y , dy 
dx^ dx 

a, b are real constants. 



^ + a— + 61, = 



18 8 Linear, nonhomogeneous 
second order equation 



a, 6 are real constants. 



SOLUTION 



■'( 



In a: - f ^-^M^1L_ + „ 
'""^ " J v{Giv)-Fm^' 

where v = xy. If G(v) = F(v), the solution is xy = c. 



Let mi,m2 be the roots of m^ + am + b = 0. Then 
there are 3 cases. 

Case 1. m,, m^ real and distinct: 

y = Cie*"i* + cge""!* 

Caae 2. m,, mg real and equal: " ' ' 

y = c,e"»i* + C2a:e"'i^ 

Case 3. rtii = p + gi, m2 — p — qv. 

y = eP*(ci cos qx + Cj sin qx) 



where p = — o/2, g = \/6 — oV4 . 



There are 3 cases corresponding to those of entry 18.7 
above. 

Case 1. 

y = Cie*"!* + Cg^"** 

mi — 7112 J 

WI2 ~ '"i ^ 
Case 2. 

+ see"!* I e~'"i^fi{x)da: 

— e*"!* I xe-'^i'^ R(x) dx 
Cases. 

y = ei*E(cj COS qx + e^ sin qx) 

+ ^— I e P- «(a:) cos oa; da; 



18.9 Euler or Cauchy equation 



S + -f + *. = S(.) 



Putting X = e', the equation becomes 
and can then be solved as in entries 18.7 and 18.8 above. 



106 



BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS 



IH^DIFFERENTIAL EQUATION 


SOLUTION 


18.10 Bessel's equation 


y = CiJ„(Xa) + C2y„(a!) 
See pages 136-137. 


x^^ + x^+ {\^x--^-n^)y = 


18.11 Transformed Bessel's equation 


y = x-r>j^e,J^,J^xA + cY^J^x^Xi 


x2 + (2p + l)a^ ^ + {a^x^^ + P^)y = 


where q = Vp^ ~ P^ • 


18.12 Legendre'a equation 


y = eiP„{x) + c^Q^lx) 
See pages 146-148. 


^^-''^^2-2x^ + ^i^ + i)y = 



IF- 




i^ ' 




RITHMETIC SERIE! 



19.1 a + ia + d) + {a + 2di + ■■■ + {a+ in-l)d} = ^n{2a + (n - l)d} ^ ^n{a-i-l) 

where 1 = a+{n — l)d is the last term. 



Some special cases are 



19.2 
19.3 



l + 2 + 3+---+n = |n(n + 1) 
1 + 3 + 6 + •■• + {2w-l) = m2 




19.4 



EOMETRIC SERIES^ 



a + ar + wfi + a-fi + • ■ • + ar"""^ 
where ? = ar«-' is the last term and r t^ 1. 




a(l - r") 

1 - r 



g — rf 
1 -r 



If -1 < r < 1, then 



19.5 



a + ar + ar^ + or^ + 



1 -r 




19.6 a + (a + d)r + (a + 2d)r2 + •-■ + {a+(n-l)d}rn-i = a(l-r«) ^ rd{l - nr"-i + (n- l)r"} 

1 - r (1 - r)2 

where r ¥^\. 



If -1 < r < 1, then 
19-7 a + (o + rf)r + (a + 2d)r2 + 



r(f 



1 - r ' (1 - r)2 



SUMS OF POWERS OF POSITIVE INTEGERS 



19.8 IP + 2" + 3P + •■■ + „p = ^^ + inp + ^^P" _ 

P + 1 2 2! 



BipW-i B2P(P - 1)(P - 2)nP-3 



4! 



+ 



where the series terminates at n^ or n according as p is odd or even, and B^ are the Bemoullx 
numbers [see page 114]. 



107 



108 



SERIES OF CONSTANTS 



Some special cases are 

19.9 1+2 + 3+ ■•■+n = "tn+D 

2 

19.10 12 + 22 + 32 + ... + n2 = Mn + l)i2n + 1> 

6 

19.11 13 + 23 + 33 + ■•■ + n3 = nHn + l)i ^ (i + 2 + 3 + • ■ • +n)2 

19.12 14 + 24 + 34 + ... + „4 = «(n + l)(2n + 1)(3«2 + 3n - 1) 

30 



If Sfc — l** + 2"= + Sif + • • • + n'^ where k and n are positive integers, then 

'^ + l\« . . /k+l 

k 



k + \ 
1 



19.13 C: ^ iSi + [■■;" )Sa +•••+[-;* )Sk = (n+ 1)^+1 - (n+l) 




19.14 
19.15 
19.16 
19.17 

19.18 
19.19 
19.20 
19.21 
19.22 
19.23 
19.24 
19.25 
19.26 

19.27 
19.28 
19.29 
19.30 
19.31 
19.32 



2^3 4^5 

3^5 7^9 

4^7 10 ^ 13 

5^9 13 ^ 17 



= ln2 



^+i 
11 14 



+ 


1 
44 


+ 


+ 


1 

48 


+ 


- 


1 
42 


+ 


- 


1 

4* 


+ 


- 


1 
46 


+ 


+ 


1 
72 


+ 


+ 


1 

74 


+ 



l_l+i_ 

2 5 8 
J- + J-+J_ 

12 ^ 22 32 

14 ^ 2^ 3« 

1+1 + X 
16 ^ 2« 3« 

1-1 + 1 

12 22 32 

1-1 + 1 

14 2* 3* 

1_1 + 1 
1« 28 38 

1 + 1 + 1 

12 ^ 32 52 

1 + 1 + 1 

1* 3* 5^ 



1+1+1 + 1 + 
16 ^ ga ^ 5« 76 ^ 

1_1+1_1+ 

13 33 53 73 

1 + 1_1_1 + 
13 ^ 33 53 73 ^ 



1*3 ^ 3-5 ^ 5-7 ^ 7-9 

^ + ^ + -^+^ + 
1'3 2-4 ^ 3-5 ^ 4-6 



rV| , V2 In <1 + V^) 

8 "^ 4 



6 

90 

_£l. 
945 

i^ 
12 

7^ 

720 

^ 31irB 

30,240 

8 

96 

960 

zi 

32 

3ir2\/2 



• ■•-■^C J' -'. .'■•■ 



1 + 1,+ 1,+ 1,+ 



16 •• -.7 . 

_ 1 
2 
3 

4 



12.32 ' 32.52 ' 52*72 ' 72*92 



16 



SERIES OF CONSTANTS 



109 



19.33 



+ 



+ 



12,22.32 22 -32 -42 32-42.52 
1 1 



+ 



47^2 - 39 



1».34 i 1 ^ , 

a a + d a + 2d a + 3d 



+ 



16 
1 ..«- 



+ 



19.35 J-+J^+J^+J_ + 

12p ^ 221' 32P ^ 42p ^ 

19.36 J-+J_+JL+J_ + 

12P ^ 32p ^ 52p ^ 72p ^ 

19.37 i--J_ + J-_J_ + 

l2p 22P S^K 42P 



X l + « 

22P-l^PBp 

(2p)! 

{22P - l),r2PBp 

2(2^)] 
(22p-i-i)^PBp 



du 
d 



{2p)\ 



19.38 



^ + 1 



12P+1 32P+I '^ 52P+I 72P+I 



+ 



;r2p + lSp 
22p + 2(2p)J 



n 



r'^A 




MISCELLANEOUS SERIES 




19.39 l + COSa + C0S2«+ ■-. + COSTla :3: ^^" ^^ ^ ^^^ 
2 2 sin (a/2) 

19.40 sin« + 3in2a + 8m3a+ ••• +sinn« = sin [^(n + !)]„ sin^n« 

sin (a/2) 



19.41 1 + rcosa + r2cos2a + r3 cos 3a + 



1 — r cos g 
X — 2r cos a + 9-2 ' 



|r| < 1 



'• .H 



19.42 rsina + r2am2a + r«sin3a + ■•• = r_^Tia ,, . 

1 -2rcosa +r2' I'^l ^ ■" 

19.43 1 + rcosa + r2cos2a + ••■ + r«coan« = r'*'^^ cos na - r" + i co3(n + l)a - r coaa + 1 

1 - 2r cos a + r2 

19.44 rsina + 7^sin2a + ■•■ + r^sinTta = ^si""-^""^^ sin(n + l)a + r"+2 8inna 

1 — 2r cos a + r^ 



hit 



THE EUIER-MACLAURIN SUMMATION FORMULA 



19.45 



n-l 

fc 



2 F{k) = f F(k) dk ~ I {F(0) + F(n)} 

+ ^{F'(w)-F'(0)} - tIo^^'"**'*"^'"^*'^ 



A. i 



+ 



+ ... (-l)p-l_^{i^(2p-I)(„) _p(2p-l)(0)) + ... 



19.46 



THE POISSON SUMMATION FORMULA 



2 F(fc) = 2 11 «27Kn«/r(a.) d:c 



{/: 



if 




20.1 



TAYiOR SERIES FOR FUNCTIONS OF ONE VARIABLE 



2! (n— 1)! " 



where R„, the remainder after « terms, is given by either of the following forms: 
20.2 Lagrange's form ie„ = /^"H$)(a; - a)" 



20.3 Cauchy's form 



_ /t">a)(x~{)"-Hx-a) 



The value $, which may be different in the two forma, lies between o and z. The result holds if f(x) has 
continuous derivatives of order n at least. 

If lim fi„ = 0, the infinite series obtained is called the Taylor series for f(x) about x = a. If 

a = the series is often called a Maclaurin aeries. These series, often called power series, generally 
converge for all values of x in some interval called the interval of convergence and diverge for all x outside 
this interval. 



BINOMIAL SERIES 



20.4 



(a+«)" = g" + na^-^x + "t"^~ ^> an-2a.2 + ^(" " IK"" 2) an-3a;3 + 



= a" + 



Special cases are 



a^-ix + 



a«-2a:2 + 



20.5 (a+a;)2 = a^ + Zaz + x^ 

20.6 {o + a;)3 - a^ + Sa^x + Saz^ + x^ 

20.7 (a + 3:)^ = a4 + 4a3a: + 6a2x2 + 4aa;3 + ar* 

20.8 (l + a:)-i = 1-x + x^-x^ + x*---- 

20.9 (l + a)-2 = 1 — 2a; + 3a;2 - 4x3 + 5a^ - • •• 

20.10 (l + a!)-3 ^ 1 - 3x + 6x2 - 10x3 + 153^ _ .. 

20.11 (l+x)-/2 = i_l^+|l|,2- 113^^3 + 



20.12 (H-x)»/2 = i+±x-^x^ + 



1-3 
2'4'6 

l-4'7 



20.13 (1 + x)— . i_l, + ll|,. ___ 



X3 - - 
X3 + 



20.14 (l + x)./3 = 1-,!, --!_,. + _il^,3_ 



(in-3a;3 + 



-1 < X < 1 

-1 < X < 1 

-1 < X < 1 

-1 < X ^ 1 

-1 < X ^ 1 

-1 < X ^ 1 

-1 < X ^ 1 



110 



TAYLOR SERIES 



111 




20,15 e' = i + x+|J + f^+ ... 



= ^n + f + T + 



-1< X < 1 



20.19 Inx = 2U^^] + ^^^ +lf?^^ll 



x+l/ '^ 3[x + lj "'■fltTf' "*" 



— « < X < * 



20.16 a- = e-in« = 1 + « ]„ a + i^il^ + t^J^L^ + . . . -« < ar < « 

20.17 ln(l + x) = a;-^ + ^-^+... -Kxfil 

23 4 i%x = j. 



20.20 ,„. = f^)+|(E^y,|(^l- 



+ 



X > 



x^i 



;eries for trigonometric functions 



20.21 sinx 

20.22 cos X 

20.23 tan z ■■ 

20.24 cot X -- 

20.25 sec x 

20.26 CSC X = 

20.27 sin- Ik 

20.28 cos~ix 






X^ . X^ 



2! ^4! 6! 



1 _ X _ x^ 2x° 
X 3 45 945 



"^ 2 "^ 24 ■*" 720 "^ 



X 6 360 15,120 "•■ 



— 00 < a; < oo 



— ~ < X < «= 



+ 1 —2 + 

(2n)! 

22nB„K2n-l 



+ 



(2n)! 



+ 



2(22''-'-l)B„x2''-' 



(2n)! 



= a: +i^+il3xf 1^3^x7 
2 3^ 2"4 5 ^ 2-4*6 7 



— jr ._, »■ / ,1x3 1-3x5, 



20.29 ten-»x = 



r 3+5 7 + 

U--i + J L + 

^2 X 3x3 5a.5 ^ 



w<| 



< |xl< »• 



w<t 



20.30 cot-'x = ^-tan-ix = 



x^ . x= 



2-(^^-T + T- ■ 

P" + x-3^+5^- 



20.31 sec-i X 

20.32 csc-ix 



2 V X 2 • 3x3 ^ 2 • 4 • 5x6 



+ 



sin-> (1/x) = - + ^— + ^'^ 

X 2 • 3x3 ^ 2 • 4 • 5x5 



+ 



< |xl < 0- 
|x| < 1 

\x\ < 1 

|x| < 1 
[+ if X £ 1, - if X ^ -1] 

lx| < 1 

[p - if X > 1, p - 1 if X < -1] 

|xt > 1 
Ixl >1 



112 



TAYLOR SERIES 




SERIES FOR HYPERBOLIC FUNCTIO 



X^ , K" 



2t ^ dt ^ fit ^ 



4! 

^"T + ir 



6! 
315 



+ 



ic 3 45 945 



x2 . 5^4 61k6 



1 - — + 

2 ^ 24 



+ 



± f In 12x1 + 



20.39 sinh-ia; = 

20.40 cosh-' X = ± -( In (2x) - 

20.41 tanh->x 

20.42 coth-»iB 



720 

31x5 
15,120 

„ _ JEL + 1-3x5 
2-3 ^ 2-4.5 



1 X 7x^ 
X 6 "^ 360 



+ 



— '^ < X < * 

— OS < X < « 
(_l)n-122n(22n_l)B^a;2t>-I 

(2^0! 

(-l)''-122«B„x2n-l 

(-1)"£„X2" 
■ (2«)I ■*" ■■■ 

(-l)«2(22n-i - l)5„x2n-i 



+ 



1 • 3 • 5x' 
2*4'6-7 

1'3 



+ 



2*2x2 
1 



+ 



(2n)! 



1'3'5 



+ 



2 • 4 • 4x4 ^ 2 • 4 • 6 • 6x8 



+ 



1*3 I l'3-6 



2-2x2 2-4'4x* 2-4-6-6xfl 



^^f^y^y^ 



a; 3x3 ^ 5a.5 ^ 73.7 



■)} [:; 



1.1 <i 



< 1x1 < 



w<i 



< |x| < a- 

lx| < L 

"+ if X ^ 1 
- if X ^ - 

if cosh-i X > 0, 
if cosh-' X < 0, 

|x|<l 
\x\ > 1 



.] 



X ^ 
X 



:a 



MISCELLANEOUS SERIES 



20.43 e»- = i+:c+f-f-^+... 



20.44 e'^°^ * 



, , x2 a:* 31x8 



20.45 efn* = 1 +x + ^+ ^ + 3£l+ ... 

^ A O 

20.46 e-sinx = x + x2+%^-^-^+ ... + 2:!^iliniW4Ix^^ ... 

a oU 90 7(,! 

20.47 e-cosx = l + x- ^-^+ ■-- + 2"^^ cos (n./4) x» ^ ... 

3 D 



n! 



on ^« 1 I • I lit ^^ a^ 3:8 

20.48 Inlamxl = m |xl - - - — - ^835 " 



20.49 inlcosxl = -^ - ^^ - g - ^^^^ 



22"-iB„x2« 
w(2m) ! "^ 

2a«-i(22«-l)g„g2" 
7i(2ti) ! 



+ 



on en 11* 1 1 1 1 . «^ . ^ar* , 62x« 

20.50 ln|tanx| = in\x\ + ^ + -^ + ^^ 



+ 



22'>(22"-i-l)B„x2" 



n(2n) ! 



20 



.51 llLikL£) = jc - (l + ^)x2 + (l + ^ + i)x8 - -■- 

X "i X 



— <* < X < <« 



— *o < X < <» 



I r ^ " 

W<2 



— =° < X < • 



— «° < X < » 



< Ixl < 



N<^ 



< kl< I 



1x1 < 1 



TAYLOR SERIES 



113 




If 
20.52 

then 
20.53 

where 
20.54 

20.55 

20.56 

20.57 

20.58 

20.59 



y = Cix + c^x^ + Cgx^ + c^^ + CsxS + c^jje + . . . 
a: = Ciw + Czv" + C^y^ + C^y* + €^y^ + Cei/S + • - • 
c^C^ = 1 

c\C^ — Bcic^e^ — 5<^ — eje4 

e^Cs = 6<!?C2C4 + 3cfc2 _ cjcj + 14c^ - 21c,c|c3 

cJ^Cfl = Tc^cgCs + 84c,c|c3 + Tcf^^^ _ 28c2c2c2 - c^^c^ - 2Sclelc^ - 42c| 



TAYIOR SERIES FOR FUNCTIONS OF TWO VARIABLES 



20.60 f(x,y) = f{a.b) + (x-a)fMb) + iy-b)f^{a.b) 

+ -^(i^- aW^ia. b) + 2(x - a){y - b)f^(a, b) + {,y- 6)»/„(o, 6)} + • • • 

where fjfl., b), fy{a, b), , . . denote partial derivatives with respect to ar, j/, . . . evaluated at x = a, y = b. 




DEFINITION OF BERNOULLI NUMBERS 



The Bernoulli numbers Bj, B^, B^, . . . are defined by the series 
21.1 '^ 



e*- 1 



= 1 _ ^ 4_ ^1*' ^2^* ^3^:8 
2'^ 2\ "IT "^ ~6r 



x\ < 27 



21.2 



, X ^x Bix^ B2X* BgajB 

1 — « cot - = — : 1- — 1 = — + 

4! ^ 6! 



2! 



I»l <^ 



DEFINITION OF EULER NUMBERS 



The Elder numbers E^,E^E^, ... are defined by the series 
21.3 s«.hx = 1-^ + ^.^ + 



2! 



4! 



6! 



W<| 



21.4 



secx 



= 1+^ + ^+^ + 



21 



4! 



6! 



W<2 



BERNOULLI AND El 



llfiRS 



Bernoulli numbers 


Euler numbers 


Bi = 1/6 


El 


= 1 


B2 = 1/30 


E2 


= 5 


B3 = 1/42 


Ez 


= 61 


B4 = 1/30 


E, 


= 1385 


B5 = 5/66 


E-, 


~ 60,521 


Bfl = 691/2730 


Ee 


= 2,702,766 


B7 = 7/6 


E, 


= 199,360,981 


Bg = 3617/510 


En 


= 19,391,512,145 


B9 = 43,867/798 


E^ 


= 2,404,879,675,441 


Bio = 174,611/330 


Eio 


= 370,371,188,237,525 


fill = 864,513/138 


En 


= 69,348,874,393,137,901 


B12 = 236,364,091/2730 


E12 


= 15,514,534,163,557,086,905 



114 



BERNOULLI AND EULER NUMBERS 



115 



JmBwW i W oi» BtU'lffC'yi'tnwB^^HiHffluwWR 



2n+ 1 
2 



2n+ 1 



21.5 ( „ ")22Bi - ^^ 4 "J24B2 + [ g ~ )2«B« - 



2n+ 1 



(_l)n-i(2n + 1)2^B„ = 2n 



21.6 



2n f/2n - 1 



*'•' ^» = ^.=(1^ |(^'"r V ^»-' - ('"3" ^"- + ('"5- ' ) ^"-3 - • • ■ (-«- 



SERIES INVOLVING BERNOULLI AND EULER NUMBERS 



21.8 B^ 



- (2n)! 



l + i + a= + 



22n-l^nl 22" 32« 



21.9 B„ = 



2(2?i) ! 



1+ 1 + 1 + 



(22" - l)n^n ] 32" 52" 



21.10 B, = 



(2n)! __L+ J_ 



21.11 £;„ = 



22n+2(2n)! K 1_ 1 

y2n+l 1^ 32n + l "•" 52n + l 




21.12 



B. ~ in^Hirei-^'^Vrn 




FORMULAS from 
VECTOR ANALYSIS 




VECTORS AND SCALARS 



Various quantities in physics such as temperature, volume and speed can be specified by a real number. 
Such quantities are called scalars. 

Other quantities such as force, velocity and momentum require for their specification a direction as 
well as magnitude. Such quantities are called vectors. A vector is represented by an arrow or directed 
line segment indicating direction. The magnitude of the vector is determined by the length of the arrow, 
using an appropriate unit. 



NOTATION FOR VECTORS 



'£ 



A vector is denoted by a bold faced letter such as A [Fig. 22-1]. The magnitude is denoted by |A| 
A. The tail end of the arrow is called the initial point while the head is called the terminal point. 



or 



2. 



3. 



FUNDAMENTAL DEFINITIONS 



1. Equality of vectors. Two vectors are equal if they have the same 
magnitude and direction. Thus A = B in Fig. 22-1. 



Multiplication of a vector by a scalar. If m is any real number 

(scalar), then mA is a vector whose magnitude is \m\ times the 
magnitude of A and whose direction is the same as or opposite 
to A according as m > or m < 0. If in = 0, then mA = ia 
called the zero or null vector. 




B. 



Fig. 22-1 



Sums of vectors. The sum or resultant of A and B is a vector C = A + B formed by placing the 
initial point of B on the terminal point of A and joining the initial point of A to the terminal point 
of B [Pig. 22-2(6)]. This definition is equivalent to the parallelogram law for vector addition as in- 
dicated in Fig. 22-2(c). The vector A-B is defined as A + {-B). 






1X6 



FORMULAS FROM VECTOR ANALYSIS 



117 



Extensions to sums of more than two vectors are immediate. Thus Fig. 22-3 shows how to obtain 
the sum E of the vectors A, B, C and D. 



(a) 



D 



Fig. 22-3 




(b) 



4. Unit vectors. A unit vector is a vector with unit magnitude. If A la a vector, then a unit vector in 

the direction of A is a = A/A where A > 0. 



LAWS OF VECTOI 



tEBRA 



If A, B, C are vectors and m, n are scalars, then 
22.1 A + B = B + A Commutative law for addition 



22.2 A + (B + C) = (A + B) + C 



Associative law for addition 



22.3 m(7iA) = {mn)A = n(m.A) Associative law for scalar multiplication 

22.4 (m + 7i)A = mA + nA Distributive law 

22.5 mi{A + B) =^ »iA + rniB Distributive law 



COMPONENTS OF A VECTOR 



A vector A can be represented with initial point at the 
origin of a rectangular coordinate system. If i, j,k are unit 
vectors in the directions of the positive x, y, z axes, then 



22.6 



A = Ax\ + AJ + A^ 



where Aii.AaJ.Agk are called ctym'poneni vectors of A in the 
i, j, k directions and A^.A^, A3 are called the components of A. 




DOT OR SCALAR PRODUCT 



Fig. 22-4 



22.7 



A • B = AB cos * ^ S S 



where e is the angle between A and B. 



118 



FORMULAS FROM VECTOR ANALYSIS 



Fundamental results are 

22.8 A*B = B-A 

22.9 A-(B + C) = A-B + A-C 

22.10 A-B = AiB^+ A2BZ + A3B3 
where A = Aii + A^ + Agk, 3 = 811 + 823 + 8^^. 



Commutative law 
Distributive law 



CROSS OR VECTOR PRODUCT 



22.11 



A X B = AB sin S u 



0^ S Sir 



where $ ia the angle between A and B and u is a 
unit vector perpendicular to the plane of A and B 
such that A, B, u form a right-handed system [i.e. a 
right-threaded screw rotated through an angle less 
than 180° from A to B will advance in the direction 
of u as in Fig. 22-5]. 

Fundamental results are 




22.12 AXB = 



i j k 

Ai Az A3 
Bi Bj ^3 




Fig. 22-5 



22.13 AXB = -BXA 

22.14 AX(B-I-C) = AXB-I-AXC 

22.15 |A X B] = area of parallelogram having sides A and B 

ISCELtANEOUS FORMULAS INVOLVING DOT AND CROSS PRODUCl 



22.16 A'(BXC) = 



Aj A2 A3 
81 82 8^ 
Ci C2 C^ 



= AjfiaCa + Aa^aCi + As^iCg - AgBgCi - AjjBiCg - AjBaCj 



22. 1 7 I A • (B X C) [ = volume of parallelepiped with sides A, B, C 

22.18 AX(BXC) = B(A>C) - C(A'B) 

22.19 (A X B) X C = B(A • C) - A(B • C) 

22.20 (A X B) • (C X D) = (A • C)(B • D) - (A • D)(B • C) 

22.21 (A X B) X (C X D) - C{A • (B X D)} - D{A • (B X C)> 

= B{A • (C X D)} - A{B • (C X D)) 



_ 



FORMULAS FROM VECTOR ANALYSIS 



119 



DERIVATIVES OF VECTORS 



The derivative of a vector function A(m) = A i(u)i + A2(u)j + A^iu^k of the scalar variable u is 
given by 

„-, dA ,. A(M + AM)-A(tt) (Ml dAa dA^ 

22.22 -J— = hm = -;— i 4- -;— j + -;— k 

du iu-*o Au du du du 

Partial derivatives of a vector function A(a;, y, z) are similarly defined. We assume that all derivatives 
exist unless otherwise specified. 



RMULAS INVOLVING OERIVATIV£$ 



22.23 #(A.B) = A.^+^-B 
du du du 



22.24 -^(AXB) = AX^ + ^XB 
du du du 



22.25 £iA.(BXC)> = ^.(BXC) + A. f XC+A.fBxf 



22.26 A.^ = A^ 
du du 



22.27 A ■ -r~ = if [A[ is a constant 



The operator del is defined by 
22.28 



THE DEL OPERATOR 



V = l~ H ]T h k-r- 

dx dy dz 



In the results below we assume that U = U{x, y, z), V = V(x, y, z), A = A(«, y, z) and B = B(a:, y, z) 
have partial derivatives. 



THE GRADIENT 



22.29 Gradient of U - grad U - 



VU = {\^ + \±.+ k^)U 
'3a: * by dz ' 



dU . ^ dU. _L 3C7. 
5a; dy dz 



THE DIVERGENCE 



22.30 Divergence of A = divA = V-A = 



's+'^ + ''i)-'^'' + ^^ + -*^'" 



SAi dAz dAs 
dx dy dz 



120 



FORMULAS FROM VECTOR ANALYSIS 



THE CURL 



22.31 Curl of A = curl A = V X A 



*l; + i^ + »'i)=^<^^*-^^^ + ^3k) 



i j k 

-1 A A 

5a; dy dz 



dy 



dz 



dA2\ fdAi SAgN fdAz dAi 



THE LAPLACIAN 



22.32 LaplacianofU = V^C/ = V'[VU) = ^+^ + ^ 

22.33 LaplacianofA = v^A = + + ^ 



THE BIHARMONIC OPERATE 



22.34 Biharmonic operator on C7 = ^*U = V^V^U) 






22.35 V(U + V) = VU + VV 

22.36 V'(A+B) = V'A+V'B 

22.37 VX(A + B) = VXA+VXB 

22.38 V-(t/A) = (VC/)'A+ [/(V'A) 

22.39 V X {UA) = iVU)XA + U(V X A) 

22.40 V'(AXB) = B'(VXA) - A-(VXB) 

22.41 Vx(AXB) = (B"V)A - B(V-A) - (A*V)B + A(V'B) 

22.42 V(A-B) = (B*V)A + (A*V)B + BX(VX A) + AX(VXB) 

22.43 V X (V IT) = 0, i.e. the curl of the gradient of t7 is zero. 

22.44 V • (V X A) = 0, i.e. the divergence of the curl of A is zero. 

22.45 V X (V X A) = V(V • A) - V2A 



FORMULAS FROM VECTOR ANALYSIS 



121 




If A(m) = ^^^(w)' *^C" t*^6 indefinite integral of A{u) is 

22.46 I A(m) du = B(it) + c c ::= constant vector 
The definite integral of A(w) from u = a to u = & in this case is given by 

22.47 r Aiu)du = B(6) - B(a) 
The definite integral can be defined as on page 94. 



IE INTI 



Consider a space curve C joining two points Pi{ai,a2,a^) and 
^2(^1. &2. ''3) as in Fig. 22-6. Divide the curve into n parts by points 
of subdivision {xi.y^.Zi), .. .,(a:„_i,y„_],z^_i). Then the line integral 
of a vector A(a:, y, z) along C is defined as 

22.48 f A-dr = r 'A-dr = lim 1 \{x„,y^,z^)' ^r^ 

where Atp - AKpi + A?/p j + Az^k, AXp = a!p+ 1 - aip, Ai/p = i/p + j - ^p, 
A«p = Zp+i — Zp and where it is assumed that as n -» «= the largest 
of the magnitudes |Arp| approaches zero. The result 22.48 is a gen- 
eralization of the ordinary definite integral [page 94]. 

The line integral 22.48 can also be written 

22.49 I A-dr = \ Aydx + A^dy + A^dz 
using A = All + Agj + Agk and dt — dxi + dyj + dzk. 




Fig. 22-6 



PROPERTIES OF IINE INTEGRALS 



22.50 



22.51 



A • dr = - J A • dr 

'-'1 P| 

rPi r^s /-pj 

I A • dr = ) A • dr + I A • dr 

-'p, *^Pj "^Pa 



NDEPENDENCE OF THE PATH 



In general a line integral has a value which depends on the particular path C joining points Pj and Fj 
in a region %. However, in case A = V^ or V X A = where <f> and its partial derivatives are con- 
tinuous in %, the line integral I A • dr is independent of the path. In such case 

22.52 r A • dr = f ' A • dr ^ 0(^2) - ^(^1) 



122 



FORMULAS FROM VECTOR ANALYSIS 



where ^(Pi) and 0(^2) denote the values of at Pi and Pa respectively. In particular if C is a closed curve, 



X 



£ 



22.53 I A'dr = * A-rfr ^ 

c *^r 

where the circle on the integral sign is used to emphasize that C is closed. 



MULTIPLE INTEGRALS 



Let F(x, y) be a function defined in a region % of the 
xy plane as in Fig. 22-7. Subdivide the region into n parts 
by lines parallel to the z and y axes as indicated. Let AAp = 
Aajp i|/p denote an area of one of these parts. Then the in- 
tegral of Fix, y) over % is defined as 



22.54 r F{x,y)dA =^ lim "2 F(Xp,yj,) AA, 

J io n-*« P=l 



provided this limit exists. 

In such case the integral can also be written as 

I F(«, y) dy dx 

i F{x,y)dy\dx 

x=a i'^v=f,(x) J 

where y = fi(x) and y - fzix) are the equations of curves PHQ and PGQ respectively and a and b are 
the X coordinates of points P and Q. The result can also be written as 



d 


y 

Aaip i!/p 


.A 


^ 


G 


r-^ 


n 


s. 




yp+i 


< 










s 


•v 




^. 










\ 


J 










% 




\q 




K 
















e 




1 
1 




^ 


H 


\ 




a 


X 


p X 


p+i 








h 



Fig. 22-7 



22.56 



I f(a:,ff)dicdi/ ^ I 1 I FKX,y)dx\dy 

« = C ^1 = 01(1*) •-'u = C L''i = gi(!() J 



where x = g^iy), x = ffziy) are the equations of curves HPG and HQG respectively and c and d are the y 
coordinates of H and G. 

These are called double integrals or area integrals. The ideas can be similarly extended to triple or 
volume integrals or to higher ■multiple integrals. 



SURFACE INTEGRALS 



Subdivide the surface S [see Fig. 22-8] into n elements of 



areaiSp, p=l,2,...,n. Let A(a;p, yp, Zp) = Ap where (Xp, j/p, 2p) 
Let Np be a unit normal to ASp at P. 



is a point P in iSp. 

the surface integral of the normal component of A 

defined as 



Then 
over S is 



22.57 



X 



A-tidS = 



lim 2 Ap 

n-*iB p=l 



N„AS„ 




Fig. 22-8 



FORMULAS FROM VECTOR ANALYSIS 



128 



RELATION BETWEEN SURFACE AND DOUBLE INTEGRALS 



If ^ is the projection of S on the xy plane, then [see Fig. 22-8] 
22.58 



Jo J J N-k 



% 



THE DIVERGENCE THEOREM 



Let £r be a closed surface bounding a region of volume V; then if N is the positive (outward drawn) 
normal and dS = N dS, we have [see Fig. 22-9] 



22.59 I VA dV = I A'dS 

The result is also called Gauss' theorem or Green's theorem. 



f V • A dF = r 

•-'it ^ e 



J 





Fig. 22-9 



Fig. 22-10 



STOKE'S THEOREM 



Let S be an open two-sided surface bounded by a closed non-intersecting curve C [simple closed curve] 
as in Fig. 22-10. Then 



22.60 4) A • dr = I {V X A) • dS 

where the circle on the integral is used to emphasize that C is closed. 



£ A'dr = r {V X A) 



GREEN'S THEOREM IN THE PLANE 



22.61 



£ Pdx + Qdy = C [^ -^\ dxdy 

J^ Jj, \3x dyj 

where R is the area bounded by the closed curve C. This result is a special case of the divergence theorem 
or Stoke's theorem. 



124 



FORMULAS FROM VECTOR ANALYSIS 



22.62 



GREEN'S FIRST IDENTITY 



J {^W + (v^)-(v^)>dv = ^{^v^) 



dS 



where p and ^ are scalar functions. 



GREEN'S SECOND IDENTITY 



22.63 



f (0VV - <pV^)dV = f (0V^ - vV0)-dS 



22.64 



X 



V X A dV = 



= 1 



(fSX A 



22.65 f ^dr = J dSXV0 



CURVILINSA,R COORDINATES 



A point P in space [see Fig. 22-11] can be lo- 
cated by rectangular coordinates {x, y, z) or curvi- 
linear coordinates (Mj, M2» "s) where the transforma- 
tion equations from one set of coordinates to the 
other are given by 

22.66 X = x{ui,U2,tts) 

y = I/{Mi,«2»"8) 

If % and Ug are constant, then as u^ varies, the 
position vector i = xi + yj + zk of F describes a 
curve called the «i coordinate curve. Similarly we 
define the 2*2 ^^d ^3 coordinate curves through P. The 
vectors dr/auj, dt/dv^ dr/du^ represent tangent vec- 
tors to the U], 112, U3 coordinate curves. Letting 
•i.«2. ©3 he unit tangent vectors to these curves, we 
have 




Fig. 22-11 



22.67 

where 
22.68 



di . dr 

dUi 



h, = 



dr 



'*■!« 


^' 31*2 


"•2=2 


dus 


^3=3 


di 
dui 


. fh = 


dr 


t ^3 ~ 


3r 

3M3 



are called scale factors. If ene^jCg are mutually perpendicular, the curvilinear coordinate system is 
called orthogonaL 



FORMULAS FROM VECTOR ANALYSIS 



1S6 



IbRMULAS INVOLVING ORTHOGONAL CURVILINEAR COORDINATES 



22.69 (£r — — ^ dui + - — du2 + T — d«3 = hi dui e, + hz du^ t^ + h.^ du^ 63 

OU| 0M2 "^Z 



d«2 = dT'di - h{dul + hldu\ + Aj**"! 



22.70 

where da is the element of arc length. 
If dV is the element of volume, then 

22.71 dV = I (/t^ei dui) • {^262 ''"2) ^ (^^3*3 ''"a) I — '^i^'^s d«i d«2 dwa 



where 
22.72 



SMi fltt2 flws 



3(a:»!/,2) 



dwi du2 dug = 



3(a:, 1/, z) 



a{Wi,U2,W3) 



duj dz£2 dug 



3(Ui, 1*2. "3) 

is called the Jaeobian of the transformation. 



Sx/dUi dx/dU2 dx/du^ 
dy/dUi dy/dii2 dy/du^ 
dz/dui dz/dll2 dz/du^ 



TRANSFORMATION OF MUlTIPlE INTEGRALS 



The result 22.72 can be used to transform multiple integrals from rectangular to curvilinear 
coordinates. For example, we have 

22.73 (((F{x.y,z)dxdydz = rff G(u,.W2.«3) J^'^'^'^W duiduzdu^ 

JJJ J J J 3(Mi,M2,M3) 

where ^C' is the region into which "3^ is mapped by the transformation and G(ui,W2»W3) ^^ ^h^ value of 
F(», y, z) corresponding to the transformation. 



0RADIENT/DIVER6ENCE, CURL ATID LAPLACIAN 

In the following, * is a scalar function and A = A,«i + Agej + AgCs a vector function of orUiogonal 
curvilinear coordinates Wi, «2> "a- 

22 JA Gradient of * = grad * = V* = T^T^+T-|^ + r-T^ 

22.75 Divergence of A = div A = V-A = i-^ U^daMi) + :S7-(Mi^2) + t?-(M2^8)1 

"in-afta L^**i °^2 <'**3 J 

hi^i ^262 ^3*3 

22.76 Curl of A = curl A = VxA = T-fr- — ~ -^ 

A1/1.2A3 dUi d«2 o«3 

hiAi h2A.2 h^A.^ 

22.77 Laplacianof* = V^* = ^^^^ ^_ ^_ — j + — (^-^ — j + — (^-^ — ^J 
Note that the biharmonic operator V** = V^iy^) can be obtained from 22.77. 



126 



FORMULAS FROM VECTOR ANALYSIS 



SPECIAL ORTHOGONAL COORDINATE SYSTEMS 



Cylindrical Coordinates (r, 6, z) [See Fig. 22-12] 



22.78 
22.79 



X = rcosff, y = r sin ff, z = z 



22.80 



V24. ^ ^+1^1+1^+^ 
3r2 ^ r Sr ^ r2 5tf2 ^ dz^ 





Fig. 22-12. Cylindrical coordinates. 



Fig. 22-13. Spherical coordinates. 



Spherical Coordinates {r,e,^) [See Fig. 22-13] 



22.81 
22.82 

22.83 



r sin e cos 0, y = r sin 9 sin <p, z = r cos e 
hi = 1, k2 = r2, fcg = r2 sin2 9 



V24. = 



r^ dr\ drJ r^ sin fl 3* 



1 324, 






Parabolic Cylindrical Coordinates {u,v,z) 



22.84 



22.85 



22.86 



— ^{u^ — v^), y = uv, z = z 



fe| = u2 + r2, fts = 1 






+ 



32* 
dz2 



The traces of the coordinate surfaces on the zy 
plane are shown in Fig. 22-14. They are confocal 
parabolas with a common axis. 







y 




u - 


=^ 


f 


11 = 


i^^i.^ 


-^ 


^ 


^^^ 


i5>^ 


.--^ 




^^--<2s 



Fig. 22-14 



FORMULAS PROM VECTOR ANALYSIS 



127 



Paraboloidal Coordinates {u,v,<i>) 



22.87 

where 
22.88 

22.89 



X = uv COS 0, J/ — Ml? sin 0, z =■ ^(u^ — v^) 
u ^ 0, u g 0, g < 2jr 



V2* = 



^ ^M^i + 



1 ALi*u 1-3^ 



«(w2 + i;2) Am I 5m y v(w2 + v2) flu I 5v / "'^'^ 9*^ 



Two sets of coordinate surfaces are obtained by revolving the parabolas of Fig. 22-14 about the 
X axis which is then relabeled the 2 axis. 



Elliptic Cylindrical Coordinates {u,v,z) 



22.90 

where 
22.91 

22.92 



X = a cosh u cos V, y = a sinh u sin v, z =^ z 
w ^ 0, ^ V < 27r, — CO < z < » 
h\ = hi = a2(sinh2M + sin2i;), hi = X 

__ 1 /a^ a^\ a^ 

~ a2(sinh2ii + sin^r) \Su^ &v^ ) dz^ 



The traces of the coordinate surfaces on the xy plane are shown in Fig. 22-15. They are con- 
focal ellipses and hyperbolas. 



Vff 



u - 2 



*1^ 



u = 1. 



V = v 


/ 


-a 


°'>» 


r 


N (fr 


a 


1" 






v = 




\ 


\ 


V/ 


1 




• 


) 


/ 




w = 2» 


\ 


^ 


<; 


// 




"u^V^ 


\ 


^ 


\ 


J 






y 


/ 


A 





u = 3/2 


s 


\ 


y 


\ 




•a 


/ 


A- 


~^ 




It = 2 


^ 


-A 


\ 


".s 


S 



Fig. 22-15. Elliptic cylindrical coordinates. 



128 



FORMULAS FROM VECTOR ANALYSIS 



Prolate Spheroidal Coordinates {$,■>], <i>) 



22.93 

where 
22.94 

22.95 



X = a sinh i sin t/ cos 0, y = a sinh £ sin 1; sin ^s, z — a cosh £ cos tj 

i ^ 0, ^ V ^ ^, O^0<27r 

kj=zhl = a2{sinh2f + sin2,j), hi - a2 sinh2 { sin* ij 



V2* = 



1 5_ / . u,5* 

a2(sinh2 ^ + sin^ 7?) sinh $ 5£ T'" a^ 



+ 



1 



a2(sinh2 { + sin^ i?) gin ij di? 



-— sin ); ^— 1 + 



32* 



dv J a2 sinh2 j sin2 i? S4>^ 



Two sets of coordinate surfaces are obtained by revolving the curves of Fig. 22-15 about the 
X axis which is relabeled the 2 axis. The third set of coordinate surfaces consists of planes passing 
through this axis. 



Oblate Spheroidal Coordinates i$,ri,4>) 



22.96 X — a cosh { cos n cos <p, y = a cosh f cos tj sin <(,, z = a sinh i sin -q 

where € - 0, -7r/2 ^ ij S ;r/2, ^ < 2;r 



22.97 



22.98 



V2* = 



hi = hi = a2(sinh2£ + sin2)7), /i| - a^ cosh^ £ cos2 , 



a2(3inh2 £ + sin2 t)) cosh f 5£ l di 



+ 



a2(sinh2 ^ + sin2 ij) cos 1? dr) 



-— cos IJ -— + 



32* 



dv / a^ C03h2 5 cos2 ,, 302 



Two sets of coordinate surfaces are obtained by revolving the curves of Fig, 22-15 about the y 
axis which is relabeled the z axis. The third set of coordinate surfaces are planes passing through 
this axis. 



22.99 

where 

or 

22.100 

22.101 
22.102 



Bipolar Coordinates {u,v,z) 



a sinh v 



y = 



a sin u 



cosh V — cos u' " cosh v — cos u ' 
^ It < 2jr, — w < V < 09, — « < 2 < » 

a;2 + (j/ — a cot w)2 = o2 csc2 u, (x — a coth ^1)2 + j/2 = *(j2 csch^ v, z = z 



h-i — h2 — 



(cosh V — cos u)2 



, /t3 - 1 



^2* = (cosh V - COS k)2 /3^ 32*\ 3^ 

a2 1 3u2 3^2; "^ 322 



The traces of the coordinate surfaces on the xy plane are shown in Pig. 22-16 below. 



FORMULAS FROM VECTOR ANALYSIS 



129 









y 


-*• 

7 


^ 


il 




ri 

V V 


*> 


\^ 




/ v = -1 A 


U 


^ 


uS 


(-0,0) or D = — ■ 


/^ 


s^y.- 


}Xpz. 




^, 




*'^ 



.e* 



(a, 0) or tf = 



<\«* 



Cv^^ 



Fig. 22-16. Bipolar coordinates. 



Toroidal Coordinates {u,v,^) 



22.103 



_ a sinh v cos _ a sinh t> sin ^ _ 

cosh V — cos u cosh V — cos u 



a sin u 



cosh V — cos w 



22.104 



' ^ (cosh u — cos u)2 ' ^ (cosh v — cos u)^ 



,2 a^ sinh^ r 
. A3 = 



22 



.105 V2* = (cosh u- cos w)^ d 



5* 



3u V cosh V — cos u du 



(cosh 1; — cos m)3 _a_ / sinh v a*\ , (cosh u — cos u)^ 3f* 
a^ sinh v Bv\ cosh v — cos u dv J (^ sinh^ v 3^2 



The coordinate surfaces are obtained by revolving the curves of Fig. 22-16 about the y axis 
which is relabeled the z axis. 



Conical Coordinates (A, p., v) 



22.106 






(^2 _ b2)(v2 - &2) 
62- a2 



22.107 



.2 _ , .2 _ \^{i.^ - V^) .2 _ X'(ji 



X2(jf2 _ ^) 



(^ - a2)(^2 - 62) 



180 



FORMULAS PROM VECTOR ANALYSIS 



Confocal Ellipsoidal Coordinates (A,^, v) 



22.108 



+ 



+ 



a2-\ ' 62 - X ' c2 - X 



3;=' 



+ 



yz 



+ 



a^ — /i b^ — fi c^ — fi 
x^ y2 z2 



- 1 X < c2 < 62 < a2 

= 1 c2<^<62<a2 

= 1 c2 < 62 < P < a2 



22.109 



fa = (g2-x)(a2-^)(aa-.) 
(a2 - 62)(a2 - c2) 

2 = (62-X)(62-^)(fr2_,) 
(62-a2){62-c2) 

^2 = (C2-X)(c2-^)(C2-.) 
(c2-a2)(c2_62) 



22.110 



hi = 



(ti - xX" - X) 



^2 



A3 



4(o2-x)(62-X)(c2-X) 

(v - fi){\ - /i) 

4(a2 - ^)(62 - /x)(c2 - n) 

(X - !■)(;. - y) 

4(a2-^)(62-^)(c2-^) 



Confocal Paraboloidal Coordinates {k,(i,v) 



22.111 



'^^ 1 


y2 


a2-X ' 


62- X 


^^ , 


1/2 


a2 — ^ 


62-;, 


«=^ . 


y^ 



= z - \ — «'<X<62 



- z - fi 62</i<a2 



a2 - p 62 - 



= Z — V 



a^ < p < ^ 



or 



22.112 



^ ^ (tl2-X)(g2-;,)(a2-,) 



yi = 



62 -a2 

(62-X){62-^)(62-p) 
a2- 62 

X + /I + F - a2 - 62 



22.113 



A? = 



1.2 
/12 



fe^ = 



(/*- 


-x)(.- 


X) 


4{(l2- 


- X){62 


-X) 


(.- 


- n)(^ - 


^> 


4(a2- 


-/*)(62 


-l^) 


(X- 


-")(/*- 


") 



16(o2 - p)(62 - y) 



DEFINITION OF A FOURIER SEtlES 



The Fourier series corresponding to a function /(x) defined in the interval c ^ x ^ c + 2L where e 
and L > are constants, is defined as 



23.1 

where 

23.2 



do , S / nnx , , rtirX 

y + 2 (a„cos-y- + 6„sin-^ 



.C + 2L 



nirx J 
coB—=—dx 



r 1 /•cfai- 



If f(x) and /'(a;) are piecewise continuous and fix) is defined by periodic extension of period 2L, i.e. 
f(x + 2L) = fix), then the series converges to fix) if a; is a point of continuity and to ^{fix + 0) + fix - 0)} 
if a; is a point of discontinuity. 



COMPLEX FORM OF FOURIER SERIES 

Assuming that the series 23.1 converges to f(x), we have 

23.3 fix) = 2 c„e'"'^^/^ 

n = — » 

where 

23.4 ''•' " xj fix)e-^'^^"^dx = V 



^K-i6„) «>0 



23.5 



tSEVAL'S IDENTITY 




lf'ifix))^ax = ^f^^lK^^l) 



GENERALIZED PARSEVAL IDENTITY 



.C + 2L 



23.6 j^ J fix) gix) dx = -g- + ^2 (anC« + K^n) 

c 

where a„,6„ and c„,d„ are the Fourier coefficients corresponding to fix) and gix) respectively. 



131 



132 



FOURIER SERIES 



iFtheTrgW^phs 





23.7 


fix) = 


{-: 


< ar < ^ 
-n- < a; < 




4 /sin a; . sin 3a; , sin 5a: , 



23.8 fix) = 1x1 = 



X < X < TT 

—X — JT < X < 



£ _ A / cosx , cosSx , cos5x , 



~2t 



fix) 



Fig. 23-1 



-2ir — n- 



Fig. 23-2 



277- 




23.9 f(x) =X, -ir<X<Tr 



2 I sinx _ sin 2a; , Bin 3x 



Fig. 23-3 




23.10 f{x) = X, < X < 2,7 



— 2 I ^'" "^ 4- sin2x , sin3x , 




23.1 1 /(x) = Isin xl, — TT < X < n- 



2 4 /cos 2x , cos4x , cos 6x 



TT TT V 1-3 3*5 5-7 




-2v —IT 



Fig. 23-5 



FOURIER SERIES 



133 



23.12 f(x) = 



sin a; < a: < - 
TT < X <2Tr 



1 1 . 2 /cos 2a: , cos 4a; , cos 6a; , 

- + - sm a; - - ( -yT^ + -gTg- + -gT,- + 



23.13 f(x) = 



cos a; < a; < :r 
— cos X —77 < X < 



8 / sin 2x 2 sin 4x 3 sin 6a; 
1-3 "^ 3-5 ^ 5-7 



23.14 fix) =a;2, -b- < a; < tt 



^ _ / cos a; _ cos2g: cos Sa; 
3 \ 12 22 "^ 32 



23.15 /{a;) = a;(ff-x), < a; < tt 



e1 _ Aqs 2a; cos4g , cos 6a; , 
T ~ ^ 12 22 32 



23.16 /(a;) - a;{ir-a;){Tr + a;), -jt < a; < tt 



, „ , sin z sin 2a; sin 3g 
1-^ ' ~^s~ 23 33 



y^ 



-2r — Jf 



/(a:) 



r^ ^ 



TT 27r 



Fig. 23-6 




Fig. 23-9 




Fig. 23-10 



134 



FOURIER SERIES 



23.17 nx) 



< a: < „-a 

1 Tr — a<X<7r + a 

r- + a<X<2n- 



a 2 /sin a COS a; sin 2a cos 2a; 



1 2 

, sin 3tt cos 3a: 



--) 



- 2o — -- 2o -^ 


-» 2a -^ — 2a - 


-3t -2.r — ff 


" ^ 2V sir 



Pig. 23-11 



23.18 f(_x) = 



x{ir — X) < X < r- 

— a;(jr — a;) — tt < a; < 



8 /sin a; sin 3a: sin 5a; , 



13 



33 



53 




Fig. 23-12 



MISCELLANEOUS FOURIER SERIES 



23.19 f{x) = ain^, —v<x<v, /i # integer 



2 sin /i^ / sinx _ 2 sin 2x 3 sin 3x 
12 _ ^2 22 - ^2 32 - ^2 



23.20 /(«) = zoBitx, —Tr<x<-tr, y.¥= integer 



2^ sin P.-W I 1 , cos a: _ cos 2a; cos 3a: 



2^2 12-^2 22 - /i2 ^ 32 _ ^2 



23.21 /(a:) = tan-i[(aBinx)/(l - acosx)], -»■ < z < ir, |a| < 1 

a sin X + -^ sin 2x + — sin 3x + • • • 



23.22 /(x) " ln(l - 2acosx + a2), _^ < 3; < ^^ |a| < 1 



—2 ( a cos X + — COS 2x + -^ cos 3x + 



23.23 /(x) = \ tan-> [(2a sin x)/(l - a2)], -n- < x < ir, |a| < 1 



a sin X + "o" sin 3x + -3- sin 5x + 



FOURIER SERIES 135 



23.24 f(x) = |tan-'[(2acosa;)/(l-a2)], -ir < x < -^r, \a\ < 1 



a cos X — -T- cos 3x + -=- cos 5x — 
S 



23.25 fix) = e>^, -^ < x < w 



2 sinh /tiT ( \_ , ■^ (— l)"(/t cos nx — n sin na:) 



23.26 f{x) = sinh /lar, -b- < a; < n- 

2 sinh /in- / sinx _ 2 ain 2x 3 sin Sa: 

12 + ^2 22 + u2 "^ 32 + u2 



23.27 /(x) = cosh iix, -tt < x < n 



2fi sinh fiTT /_1 cos a: cos 2g _ cos 3g , 

TT I 2/x2 12 -|_ ^2 "^ 22 + /i2 32 + ^2 



23.28 /(x) - In jsin^x], < a; < n- 



-(1„2+ c^+C0|2£^C0|3x^ 



23.29 fix] = ln|cos^K|, -w < x < v 



_ f 1 2 — <^osg COS 2a: _ cos 3x , 



23.30 fix) = ^5r2 - ^irx + ^2:2, ^ a: ^ 2)r 



cos a: cos 2a: , cos3g , 
12 "*■ 22 32 



23.31 /(x) ^ ^X(X - n){x - 2;r), g X S 2;r 



sinx sin2x ainSx , 

13 ■•" 23 33 



23.32 fix) = ^^^ - ^w^^ + ^7rx3 - ix4, ^ X S 2. 



cos a; cos2x cos3x , 
14 "*■ 24 3^ 




^^^^^^^^^^^^^^^^ BESSEl'S DIFFERENTIAl EQUATION 

24.1 x^y" + xy' + {x^-nZ)y = n^O 

Solutions of this equation are called Bessel functions of order n. 



BESSEL FUNCTIONS OF THE FIRST KIND OF ORDER n 



24.2 



24.3 



24.4 



Jn(x) = 



X" 



1 - 



+ 



2«r(n + l)l 2(2m + 2) 2 • 4{2n + 2)(2n + 4) 



^ "I (-l)''(a;/2)" + 2fe 



J-nM = T^z: 



1 - 



+ 



X^ 



2-n r(i _ „) Y 2(2 - 2n) 2 • 4(2 - 2n)(4 - 2n) 

^ (-l)fc(g/2)2fe-" 
k=o /c ! r(fc + 1 - ?i) 

J_„(x) = i-l)^J„(x) n = 0,1,2.... 



If n#0,l,2,..., y„(x) and J_„(x) are linearly independent. 

If 717^0,1,2,..., J„ (a:) is bounded at x =^ while ./_„(x) is unbounded. 

For n = 0, 1 we have 

24.5 Jo{x) — 1 — 22 + 22. 42 ~ 2^ * 4^ * 6^ + " ' ' 

_ . , - , . _ » _ g^ , ac^ x[ 

24.0 Ji(a;) - ^ 22 • 4 "*" 22 • 42 • 6 22 • 42 ■ 6^ • 8 

24.7 /o'C'^) = -M'^) 



BESSEL FUNCTIOI 



^KIND OF ORDER n 



24.8 



i'nla:) = 



/„(«;) coswn- — ./-n(a:) 



sinTtTT 



Jp(x) cospn- - ./-p(a:) 
lim -i- 



M9^0,l,2, .. 



n = 0,1,2, .. 



This is also called Weber's function or Neumann's function [aUo denoted by iV„(a;)]. 



1S6 



BESSEL FUNCTIONS 



137 



For n = 0, 1, 2, . . . , L'Hospital's rule yields 

24.9 Y^ix) = -{ln{x/2) + y}J„{x) -- '^ in-k-l)\ix/2)^k-n 

T IT fc=o 

--2 (-iH-i>(fe) + f(n+fc)} 7;'^^ 

^ fc=o kl{n + k)\ 

where y = .5772156. , . is Euler's constant [page 1] and 

24.10 «i.(p) = 1 + 1+ 1+ ••• +-, <i.(0) = ■ 
For n = 0, 

24.11 Y,{x) = |on(x/2) + r}Jo(x) + f|f-^(l + i) + 2^{l + i + i) 

24.12 r_„(x) = (-l)''Y„(x) n = 0,1,2,... 
For any value ti £ 0, J nix) is bounded at a: = while 3'„(a;) is unbounded. 




24.13 y = AJJx) + BJ.^ix) 

24.14 y = AJJx) + BY„[x) 



n-^ 0,1,2, 
all n 



/ax 
where A and B are arbitrary constants. 



GENERATING FUNCTION FOR 3n{z) 



24.16 



gX(t 



-i/t)/2 = 2 J„(a;)(« 




24.17 
24.18 
24.19 
24.20 



RECURRENCE FORMULAS FOR BESSEL FUNCTIONS 



■'n + l(«) = —•'«(«) - •^n-l(a;) 



a;J4{a!) ^ a;/„_j(x) — nJ„{a;) 
a;J^(a;) = nJ'„(ir) - xJ^^-^{x) 
d 




dx 



24.21 

24.22 

The functions Y„{x) satisfy identical relations. 



{x»J„(a:)) = os»J^-i{x) 



^{x-«J^{x)} = -x-r'J^^.ix) 



138 



BESSEL FUNCTIONS 



BESSEt FUNCTIONS OF ORDER EQUAL TO HALF AN ODD INTEGER 



In this case the functions are expressible in terms of sines and cosines. 



24.23 Ju2(x) = \-^sin!t 



2 . 



24.26 J_,Mx) = -W-^ 



2 /cos X 



WX \ X 



+ sin a: 



24.24 J_,/2(a:) = V— cos* 

vx 



24.27 75/2(3 



— 1 1 sin X — — cos X 

X 



24.25 J,„(x) = ylJ(^^-cosx^ 24.28 J.,„M - ^/^jf sinx + (|- 1 

For further results use the recurrence formula. Results for yi/2(«). Y^/si^), . . . are obtained from 24.8 



HANKEL FUNCTIONS OF FIRST AND SECOND KINDS OF ORDER n 



24.29 Hl,'\x) - J,ix) + iY,{x) 



24.30 Hl^'ix) - J«(x) -ir„(x) 



BESSEL'S MODIFIED DIFFERENTIAL EQUATION 



24.31 . x^" + xy' - {x^ + n^)v = n^O 

Solutions of this equation are called modified Bessel functions of order n. 



lED BESSEL FUNCTIONS OF Tl 



24.32 /„(») = i-^'JJix) = e-^^!^J^(ix) 



JT" 



1 + 



+ 



xi 



2"r(« + l)r 2(2?i + 2) 2 • 4{2n + 2)(2n + 4) 



KIND OF ORD1 



+ -■ - 2 



24.33 



/_„(x) = i"J-„(ix) = e^^'^J.^iix) 



a;-n 



1 + 



+ 



X4 



+ 



2-«ra-n)\ 2{2-2n) 2*4{2-2n)(4-2n) 

24.34 I-^{x) = 7„(x) n = 0,l,2,... 

If n -/■ 0,1,2, .. ., then /„(«) and /_„(a!:) are linearly independent. 



For n = 0, 1, we have 

^ 4. ^ A- »° 

22 "^ 22>42 "*" 22 '42. 62 



24.35 /o(x) = 1 + 22 + 22712 + 02. d2.,-- + 



•9A '«A r / ^ — £ J. "^ 4. g° , x'' , 

^*».-*o 7i(a;j - 2 "*" 22.4 22 -42 '6 2^ • 42 • 62 • 8 



{a:/2) 



N + 2h 



ft=ofc!r(n + Jfc+l) 



= 2 



(a:/2) 



2fc-« 



fc=o k\T{k + l-n) 



24.37 /^(x) - /i(x) 



BESSEL FUNCTIONS 



139 



MODIFIED BESSEL FUNCTIONS OF THE SECOND KIND OF ORDER n 



24.38 K„(x) = 



—^— {/_„(x) - /„{x)} « ^ 0, 1, 2, . . . 

2 sin njT 



Urn 



{/_„(x)-/»} n = 0,1,2,... 



p^n 2 sin Pit '' ^ 

For n = 0, 1, 2, . . . , L'Hospitars rule yields 
24.39 KJ,x) = (-1)"+ Hln (a:/2) + 7>/„(a;) + J- 2 ("DM^ - A: - 1) ! (a:/2)2k-« 



«-i 
2 2 



+ 






2 k^ofc!(« + ft)! 



{*{fc) + ^(n + fc)} 



where *(p) is given by 24.10. 
For n = 0, 

24.40 ifoCx) = -{In (x/2) + y}/o(«) + ^ + 3^742(1 + *> + gg.^g.ga (1 + i + i) + " ' * 

24.41 K-J?c) = K^{x) 71 = 0,1,2,... 



X* 



GENERAL SOLUTION OF BESSEL'S MODIFIED EQUATION 



24.42 

24.43 

24.44 

where A and B are arbitrary constants. 



V = Al^ix) + Bl.^ix) 


n^0,l,2 


y = AI„(x) + BK^ix) 


all n 


y = Al^ix) + BI^(x) f-^ 
^ xl„{x) 


alln 



GENERATING FUNCTION FOR /n(x) 



24.45 



ei(t + i/«/2 = 2 /„(a:)t" 



RECURRENCE FORMULAS FOR MODIFIED BESSEL FUNCTIONS 



0-, 

24.46 /,+ ,(x) = I„-i(x) ~ —I„(x) 

24.47 lUx) = ^{/„„,(x) + /„+i(x)} 

24.48 x/;(a!) = xl^-iix) - nl„(x) 

24.49 xl'„(x) = xl„^i{x) + nl„(x) 



24.50 ^{x^Ux)} - x"/«_i(a;) 



277 

24.52 K^+.ix) = A-„_i(x) + — K«(ar) 

24.53 <(a;) = ^iK,- lix) + K^^t(x)} 

24.54 a:<(x) = -xK^.^ix) - nK^ix) 

24.55 xK;(x) = ni:„(x) - xK„+i(«) 



24.56 ^{x"£:„(x)} = -x-iC„_i(a:) 



24.51 ^{a:-n/,(x)} = x-«7„ + i(x) 



24.57 ^{x-"iC„{x)} = -x-"K,+ ,(x) 



140 



BESSEL FUNCTIONS 



MODIFIED BESSEL FUNCTIONS OF ORDER EQUAL TO HALF AN ODD INTEGER 



In this case the functions are expressible in terms of hyperbolic sines and cosines. 



24.58 /i/2(x) = -\|— sinha; 



24.59 /_i/2(a:) = A/4co8ha: 



24.61 /_3,,(x) = ^(^sinh.-^ 



24.62 /5/2(a:) - a/— "1(4+0 sinhx - - cosh a; 



24.60 /3/2(x) - A/-±-fcoshx- 



sinha; 



•\ / — 1 ( ~9 + 1 1 co3^ ^ sinh X > 



24.63 /_5/2(x) = ^/^■{(^+ 11 cosh: 



For further results use the recurrence formula 24.46. Results for K^/^i.^), K^/2{x), ... are obtained 
from 24.38. 



Ber AND Bei FUNCTIONS 

The real and imaginary parts of Jjt (aJe3»ri/4) are denoted by Ber„ (a;) and Bei„ (x) where 
OAAA x» / ^ -V (a;/2p + « (3n + 2fe)rr 



24.65 

If « = 0, 
24.66 

24.67 



r, ■ / \ X (a;/2)2fc + " . (3n + 2fc):r 



Ber(a:) - 1 " ^72" + -4T2 •' 

Bei (x) = (a:/2)2 - -i-g^ + ^,^ 



Ker AND Kei FUNCTIONS 



The real and imaginary parts of e~'^'^^^ KJ^{xe^''^) are denoted by Ker„ (a;) and Kei„ (x) where 



24.68 



Ker„ (a:) = -{In (a;/2) + y} Ber„ (x) + ^^ Bei„ (a:) 






(k - fc - 1) ! (^/2)2fc-" (3n + 2A:);r 

r-, cos ;; 

k\ 4 

1 S^ (a;/2)«+2fc 



(3n + 2kW 
4 



24.69 



Kei„ (a:) = -{In (x/2) + y) Bei„ {x) - J,r Ber„ (x) 



n-l 






(TO-A:-l)!(a;/2p-« . (Sti + 2fc)7r 



. {3n + 2fc)7r 



_^ 1 ^ a;/2« + 2ic , ^, _^ , „ . (3n + 



and * is given by 24.10, page 137. 
If 71 3:: 0, 



(a;/2)'' 



(a;/2)8 



24.70 Ker (x) = -{In (x/2) + y} Ber (x) 4- jBei (x) + 1 - -^^{1 + ^) + ^^(1 + ^ + ^ + J) - 



24.71 Kei (X) = -{In (x/2) + y} Bei (x) - ^ Ber (x) + (x/2)2 - ^^ d + ^ + i) + 



BESSEL FUNCTIONS 



141 



DIFFEI 



"fOR Befe^fKer, Kei FUNCTfONS 



24.72 



kV + xy' ~~ {ix^ + v?)y = 



The general solution of this equation is 
24.73 y = A{Ber„ (a;) + i Bei„ (x)} + B{Ker„ [x) + i Kei„ (a:)} 

GRAPHS OF BESSEL rUNCTIONS 





Fig. 24-1 



Fig. 24-2 




1— a; 




Fig. 24-3 



Fig. 24-4 



Beia 




Kei X 




Fig. 24-5 



Fig. 24-6 



142 



BESSEL FUNCTIONS 



INDEFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS 

24.74 C xJQ(x)dx = zJi {x) 

24.75 C x^Joix) dx = x^Jiix) + xJo(x) - f Joix) dx 

24.76 Cx-^jQ{x)dx = x'^Jiix) + (m-l)x'"-^ Jo(x) - (m-l)2 f x'»-2 ^(a;) 



dx 



M^) 



Mx) 






)dx 



24.77 

24 78 f ^ - 



J„(x) 



dx = 



Jy{x) 



Joix) 



-1 (?n-l)2j 



Jai'^) 



dx 



24.79 r^i(a;)da = -Jo{x) 

24.80 ra:yi(a:)da; = -xJo(x)+('jQix)dx 

24.81 ra;'"Ji(a:)da: = -x« /„ (*) + *» f ^"""^ *'o(») (^a^ 

24.82 r^^dx = -J, (x) + J Jo (a') da: 



y,(x) 



24.83 f-'-l^dx ^ -Jl^^^^r-^dx 
J x^ 7nx"*~^ m ./ a;*" * 



24.84 fx^Jn-iix)dx = x^Jnix) 

24.85 ( x-^J„+i(x)dx = -x-^J„{x) 
C x'^J„{x)dx = -x'^J„-i(x) + (m + n-1) J z"""' J„_i(x) d« 



24.86 



24.87 



C xJn{ax)Jn(px) dx = 



x{a JJPX) Kiax) - /3 J^(aX) J^M) 



)S2~ a2 

^2 / «2 



24.88 Ja;/2(«x)dx = |- V;(««)}2 + y (^1 " ^j(-'n('«=^)P 

The above results also hold if we replace J„{x) by Y„{x) or, more generally, A y„(x) + B y„(a!) where 



A and £ are constants. 



DEFINITE INTEGRAiS INVOLVING BESSEL FUNCTIONS 



24.89 Ce-'"Mbx)dx = 

24.90 C e-'"J^{bx)dx = 

24.91 I cos ax Jo(bx) dx 



Va2 + &2 



(Vgg + 62 - a)n 

6"Va2 + 62 

1 



Va2-62 




n > -1 

a> 6 
a< 6 



BESSEL FUNCTIONS 



143 



24.92 C J„ibz)dx =1 n>~l 

24.93 J^ ^^^^ = i n = l,2,3,... 

24.94 r e-'^jQ{by/^)dx = ?lU^ 
Jq a 

24.95 I X Jn(ax) J^ifix) dx = 



i82-a2 



24.96 C xJliax)dx = ^U;(«))2 + ^{l-n2/«2){Jn(«)P 
J a:yo(o3;)/o(j3fl:) (ix = ^^^ 



24.97 



24.98 yo(a:) 

24.99 ^.(x) 

24.100 ^(x) 

24.101 Yo(x) 

24.102 /o{:c) 



^2 

INTEGRAL REPRESENTATIONS FOR BESSEL FUNCTIONS 
= - I cos {x sin 9) d9 

= — I cos (n* — ac ain tf) d*, n = integer 

= _ I COB (a; giQ g\ cos^ tf dtf, Tt > —4 

2 r" 

= I COS (a: cosh w) du 



- I cosh {x sin e) d$ = ~ \ e* ^me d* 

ttJ,, 2:rJ„ 



ASYMPTOTIC EXPANSIONS 



24.103 J„(rr) 

24.104 Y,{x) 

24.105 J,(:r) 

24.106 Y^{x) 

24.107 /„(») 

24.108 £:„(ie) 



2 / na- _ 

COS ar — s" ~ T 
jra; \ 2 4 



2 . / nr- jr 

Sin I X ~ 

TTX \ 2 



'2s^ \2.nJ 

2_! ex 
n-n V 2n 



V2^ 



where z is large 

where x is large 

where n is large 

where n is large 
where x is large 
where x is large 



144 



BESSEL FUNCTIONS 



ORTHOGONAL SERIES OF BESSEL FUNCTIONS 



Let Xi, \2> M,--- be the positive roots of R J„ (x) + Sx J^(x) = 0, n> -1. Then the following aeries 
expansions hold under the conditions indicated. 



S ~0, R^'O, i.e. Ai, A2, As, . . . are positive roots of Jn(x) = 



24.109 

where 
24.110 



fix) = A^J^iXix) + A2Jni\2^) + A^J^iX^x) + 
2 C^ 



In particular if n. = 0, 

where 
24.112 



2 r' 

^fc ^ 72777 I a;/(a;)^o(M)d:a' 



RIS > -« 



24.113 

where 
24.114 



/(jr) = A^J^{\^x) + A2J„(X2a;) + A3J„(X3a:) + 



A^ = 



J */(a:).^n('^fca;)dx 



In particular if m = 0, 
where 



R/S = -n 



24.117 

where 



24.118 



24.119 

where 



24.120 



fix) = AqX» + AiJ^{\ix) + AzJniH'^) + ••• 
Aq = 2(»+l) r x^+^fix)dx 

"^^ " j''(x)-j ^xw — ;r; J <^m'^niK^)dx 

^ •'n(Xfc) — J„_i(Xfc) J„ + i{Xk) -'o 

In particular if n = so that i2 = [i.e. Xi, Xg, X3, . . . are the positive roots of ^i (a;) = 0], 
fix) = Ao + AiJoiXiX) + AzJoiXzx) + ••■ 

Ao = 2J xf(x) 



dx 



A^ = 



4iw 



J X fix) J^iX^x) dz 



BESSEL FUNCTIONS 



145 



R/S < -n 



In this case there are two pure imaginary roota —iKq as well as the positive roots \i, \2> H' 
and we have 



24.121 

where 

24.122 



fix) = Ao/„{Xox) + AjJ„(Xia:) + AsJniM^) + •" 

j X fix) J„(\kX) dx 



A. = 



Jni\k) - Jn-liK)Jn-niW ^0 



M^^UANEOUS^^^^ 



24.123 cos (a; sin «) = Jo{x) 4- 2Ji{x)cos2e + 2Jt{x)cos4e + ■•■ 

24.124 sin{a;sinfl) = 2Ji(x)ame + 2J3(x)sin3ff + 2J5(x)sin5tf + ■•• 

24.125 Jni^ + y) = 2 Ji,ix)Jr,-k{y) n = 0, ±1,^2, . .. 

fc= -" 

This is called the addition formula for Bessel functions. 

24.126 1 = Jo{x) + 2J2{x) + ■■• + 2/2«(«) + ■" 

24.127 X = 2{Mx) + ZJsix) + 6J5(x) + •■■ + (2n+l)J2„+i(x) + ••■) 

24.128 a:2 = 2{4J2ix) + ieJ^(x) + 36Je(a:) + •■■ + (2n)2/2„(a;) + •••} 
X J I (x) 



24.129 



= J^ix) - 2Mx) + SJ^ix) - ••■ 



24.130 1 = 4(x) + 2Jlix) + 2JI(x) + 2j|(ar) + ••• 

24.131 J'^ix) = i{J„-2(x) - 2J„{x) + ^„ + 2(«)> 

24.132 J'^'ix) = iU„-8(«) - 3J„_x(ac) + 3J„ + j(x) - Jn + 3(a:)} 
Formulas 24.131 and 24.132 can be generalized. 

24.133 K{x)j^^ix)-JL,JM = ^''""' 



vx 



24.134 y„(x)J_„ + i(x) + y-„(a:)J„_i(x) = ^ "" "'^ 



24.135 J„^,{x)Y„{x) - JAx)Y,^tix) = - 

24.136 sin a; = 2{J,(x) - J^ix) + ^{x) - •■•} 

24.137 cosx = Jo(x) - 2J2(a=) + 2^4(2:) - ••■ 

24. 1 38 sinh x =^ 2{/,(x) + h{x) + h{x) + • ■ ■} 

24.139 cosh x = /©(x) + 2{/2(x) + I^ix) + h(x) + • • •> 



LEGENDRE'S DIFFERENTIAL EQUATION 



25.1 (i^j^2)y" _ 2xy' + n(n + l)y = 

Solutions of this equation are called Legendre functions of order n. 



LEGENDRE POLYNOMIALS 




If m — 0, 1, 2, . . ., solutions of 25.1 are Legendre polynomials Pn(a:) given by Rodrigue'a formula 



25.2 



^'^(=^> = 24!£^(^^-^>" 



^ 



25.3 P^ix) 

25.4 P,{x) 

25.5 P^{x) 

25.6 P^ix) 



^ 1 



1(3x2-1) 
^(5x3 - 3x) 



SPECIAL LEGENDRE POLYNOMIAL 

25.7 P^{x) = ^(35x^-30x2+3) 

25.8 P^ix) = ^(63x5 - 70x3 + 15a;) 

25.9 Ps(x) = Jg{231a:8-315x-» + 105x2-5) 

25.10 Pj(x) = J^(429x7- 693x5 + 315x3 -35x) 




LEGENDRE POLYNOMIALS IN TERMS Of ^ WHER! 



25.11 Fo(cos#) 

25.12 P,(cos(?) 

25.13 /»2(costf) 

25.17 Pe(coss) 

25.18 PyCcoss) 



1 25.14 P3(cos«) = ^(3 cos tf + 5 cos 3?) 

costf 25.15 P4(costf) = JL(9 + 20cos2ff + 35 cos4ff) 

J(l + 3 cos 29) 25.16 Pstcos 8) = i|8(30 cos « + 35 cos 3a + 63 cos 59) 



_ 1 



~ ^(50 + 105 cos2ff 4- 126 cos4ff + 231 cos 6ff) 

- 15^(1 "^B cos 9 + 189 cos 3a + 231 cos 5# + 429 cos 7«) 



25.19 



GENERATING FUNCTION FOR LEGENDRE POLYNOMI 



= 2 Pn(^)t" 



Vl - 2tx + (2 n=0 



146 



LEGENDRE FUNCTIONS 



147 



25.20 
25.21 
25.22 
25.23 
25.24 



URRENCE FORMULAS FOR LEGENDRE POLYNOMI 



(n + l)P„+,(a!) - (27i+l)xi>„(x) + nP^_,{x) 

F; + i(x) - xP:(x) = (n+l)P„(x) 

xP;(x) - P'„-iix) = nP„(x) 

K + A') - P'n-iix) = (2n+l)P„ix) 

(x^~l)P;,{x) = nxP^(x) ~ nP^_,(x) 



^ 




25.25 



25.26 



OLYNOMI 



P^{x)P^{x)dx = m^n 



r, 

J_ {Pnix)} 



^dx = 



2n+ 1 
Because of 25.25, Fm(«) and P„ix) are called orthogonal in -1 ^ a; ^ 1. 



25.27 

where 
25.28 



jOGONAl SERIES OF LEGENDRE POLYNO 

/(x) ^ A^P^ix) + A,P,(x) + A^Pzix) + ■■- 
2k +1 C^ 



A^ - 



J mP^(x)dz 



iSPECIAL RESULTS INVOLVING LEGENDRE POLYNOMI 



25.29 P„(i) = I 
25.32 

25.33 

25.34 
25.35 
25.36 



25.30 P„(-i) = (-i)t 




PniO) = i 



25.31 P„(-x) = (-l)"P„(x) 
n odd 



(-1) 



n/2 



1 ■ 3 ■ 5 • ■ • (71 - 1) 



n even 



1 /"^ 

''nCa;) = - I (« + Vk^-I cos 0)" d^ 



/^n(a: 



)da; = 



^«+i{»)-P„-i(x) 



2n + l 



-"W = 2^£#^''^ 



where C is a simple closed curve having x as interior point 



148 



LEGENDRE FUNCTIONS 



GENERAL SOLUTION OF LEGGNDRE'S EQUATION 

The general solution of Legendre's equation 13 
25-37 y = A[7„(x) + BV^{x) 

where 

25.38 U ix) = 1 - ^^^^±11x2 + "(»-2)(^ + l)(^ + 3) _ ^ , 

n\ / 2! 4! 

25.39 VM = X- t"-l)(^ + 2) (n-l)(n-3)(n + 2)(« + 4) 

n\ / 3! SI * 



These series converge for — 1 < x < 1. 




LEGENDRE FUNCTIONS OF THE SECOND KIND 




25.40 

where 
25.41 



If n = 0, 1,2, . . . one of the series 25.38, 25.39 terminates. In such cases, 

C/„(x)/f;„(l) n = 0,2,4,... 
'f'n(«)/^n(l) »i -1,3,5,... 
n2 / 



C;„(l) = (-l)"/2 2n 



ir- 



n\ 



n = 0,2,4, ... 



25.42 



^n{l) = (-l)<''-l)/2 2"-» 



n-1 



nl n= 1,3,5,... 



The nonterminating series in such case with a suitable multiplicative constant is denoted by Q„(x) and 
is called Legendre's furiction of the second kind of order n. We define 



25.43 



Qnix) = 



' U„{l)V„ix) n = 0,2,4,... 
~VM)U„ix) M-1,3,5,... 




25.44 



25.45 



25.46 



lAl LEGENDRE FUNCTIONS OF THE SECOND KIND 



Q.(x) = |ln(i±f 



<?'<^> = p4\H^-^ 



nu\ - 3x2-1 / l + x \ 3x 




25.47 



i^\ - 5a3-3x, f l + x \ 5a:2 , 2 



The functions Q„(fl:) satisfy recurrence formulas exactly analogous to 25.20 through 25.24. 
Using these, the general solution of Legendre's equation can also be written 



25.48 



y = AP^ix) + BQ^ix) 



26.1 



LEGENORE'S ASSOCIATED DIFFERENTIAL EQUATION 



(1 - x'^)y" - 2xy' + ^ n(n + 1) - 



l-a;2 



y ^ 



Solutions of this equation are called associated Legendre functions. We restrict ourselves to the im- 
portant case where m, n are nonnegative integers. 



■GENDRE FUNCTIONS Qd 



26.2 



rim 
P^(X) = (1- 0:2)^/2 ^P„(X) = 



(1 _ x2)m/2 (fm+n 



where Pni^) are Legendre polynomials [page 146]. We have 
26.3 Plix) = PJx) 



-(a.2-1). 



26.4 



P^{x) = if m > n 



SPECIAL ASSOCIATED LEGENDRE FUNCTIONS OF THI F|RST KIND 



26.5 Plix) = (l-a;2)i/2 



26.6 PUx) = 3a;(l-x2)i/2 



26.7 PUx) = 3(1 -ir2) 



26.8 Plix) 

26.9 Plix) 

26.10 Plix) 



|(5a;2-l)(l-a;2)i/2 

15a;(l-x2) 
15(l-a;2)3/2 




26.11 



GENERATING FUNCTIO^O^S 



(2m)!(l-gg)"'^gt" 



2 i'r(a')i" 
n=in 



26.12 
26.13 



RECURRENCE FORMUL 




{n + l-m)P'^+iix) - (2n + l)xP';:(x) + in + 7n)P^-i{x) = 
Pr\^)-W^^zPr\^) + (n-m)in + m + l)P:ix) = 



(1 - a;2)i 



149 



150 



ASSOCIATED LEGENDRE FUNCTIONS 




26.14 



26.15 



f P™{a;)Pr{x)dx = if n^Z 



'•'-1 



)}2da! = 



2 (m + m) ! 



2n + 1 (rt - Tn) ! 




26.16 

where 
26.17 



THOGONAL SERIES 



fix) = A^PZix) + A^^.PZ^iix) + A^^^Pl+^M + 



A. = 



2k + 1 {k-m)] 



^ - ~l~(itT^ J_/<^'^'=t^''^^ 



i£. 



ASSOCIATED LEGENDRE FUNCTIONS OF THE SECOND KIND 



26.18 



(fm 



where Q„{x) are Legendre functions of the second kind [page 148]. 

These functions are unbounded at a: = ±1, whereas P^(x) are bounded at a: — ±1. 
The functions Q^ix) satisfy the same recurrence relations as P!^{x) [see 26.12 and 26.13]. 



GENEJ 



IF LEGENDRE'S ASSOCJ 



kOUATION 



26.19 



V = AP:^(x) + BQr(a') 



17 



HERMrTE POLYNOMIALS 




27.1 



NTIAI EQUATI 




y" — 2xy' + 2ny = 




If n = 0, 1, 2, . . 
Rodrigue's formula 

27,2 



RMITE POlYNOMIAtS 




then solutions of Hermite's equation are Hermite polynomials H^(x) given by 



H^x) = (-i)nex'£L(g-^^j 




27.3 


Hoix) = 1 


27.4 


Hi(x) = 2x 


27.5 


Hsix) = 4a;2 - 2 


27.6 


Hsix) = Sx^- 12a; 



P6CIAI HERllTri»OlYNOMIALS 




27.7 Hi(x) = 16a;-i - 48z^ + 12 

27.8 H^{x) = B2z^ - 160x3 + i20x 

27.9 H^ix) = 64fl:8 _ 480x4 + 720x2 - 120 

27.10 H^(x) = 128x' - 1344x5 + 3360x3 - 1680^ 




27.11 



OENERAtiNG FUNCTION 



,2tz-t' ^ -V ^n(a^) *" 




nio nl 



27.12 
27.13 



EECURRENCE FORMUL 



■«n+i('K) = 2xff„(x) - 2nH„_i(x) 
Hl,ix) = 2nH„_j{z) 




151 



152 



HERMITE POLYNOMIALS 



27.14 



27.15 



ORTHOGONALITY OF HERMITE POLYNOMIALS 

r 6-==* H^{x) Hnix) dx = m¥=n 



27.16 

where 
27.17 



ORTHOGONAL SERIES 



fix) = A^H^ix) + A,H,{x) + A^H^ix) + 



A^ - ^= f e'^' f{x)Ht,ix)dx 



27.18 



SPECIAL RESULTS 



BJ,) = (2x)n - 2l(ZL^(2.)n-:. + ^(^- l)(^^;2)(--3) (2.)n-4 , 



27.19 H„(-a;) = (-l)''fl„(a:) 27.20 H2„_,(0) = 

■ff2n(0) = (-1)''2« • 1 • 3 • 5 ■ ■ • (2m,- 1) 
«„ + ,(«) H„+i{0) 



27.21 
27.22 



27.23 



27.24 



27.25 



J ^«(t) 



±f.-^ 



dt = 



2(n + 1) 2{n + 1) 



,-!* 



dx 



{e-'^H^ix)} = -e-^fl„ + i(a;) 



J ("e"** Hnixt) dt = Vvnl P„(a;) 



27.26 ^ i /n 

fc=0 

This is called the addition formula for Hermite polynomials. 



HJ^x^y) = 2 ^j^(l]H^{xyf2.)H^.^(yyf2) 



27.27 



" H^{x)H^{y) _ g„+,(ir)H„(y) - H„(x)H^+,iy) 



2" + %! (a — 1/) 



28.1 



xy" + {X-x)y' + ny = 





lAOUERRE POLYNOMIALS 



Zl7J:\-l:J. '*"'" '°'''**°"' '' Laguerre's equation are Laguerre polynomials L„ix) and 



by Rodrigue'a formula. 
28.2 



are given 



rf« 



^» = ^:^i^"'-') 



da:" 




28.3 


Lo(x) 


= 1 


28.4 


L,{x) 


= ~x+l 


28.5 


L^ix) 


= 3:2 - 4a; + 2 


28.9 


L^{x) 


= x6 - 36*5 + 


28.10 


Liix) 


= -a;7 + 49a;fl 



SPECIAL LA6UERRE POLYNOMIAL! 



28.6 Lsix) = ~3^ + 9x2 - 18a; + 6 

28.7 ^^(a;) ~ x^- 16x3 + 72^2 - 96a; + 24 

28.8 Lsix) = -x^ + 25x4 - 200xS + 600x2 - 600« + 120 
x6 - 36x5 4- 450x4 _ 2400x3 + 5400x2 - 4320x + 720 
-x7 + 49xfl - 882a:5 + 7360x4 - 29.400x3 + 52,920x2 - 36,280x + 5040 




28.11 



GENERATING FUNCTION 



- L„{x) <n 




e-xtn~t 
1 - t 




28.12 
28.13 
28.14 



K+iix) - (2n + 1 - x) L„(x) + m2L„_,(x) = 

lUx) ~ nLl_i(x) + nL„_^(x) = 

a!Z*^(x) = nL^ix) - n2L„_i(x) 



153 



154 



LAGUERRE POLYNOMIALS 




28.15 

28.16 



MlAlS 






28.17 

where 
28.18 



ORTHOGONAL SERIES 



f(x) = AoL(,(x) + A^Liix) + A^Lzix) + 



1 r°* 



28.19 L„(0) = m! 



28.21 



28.22 



28.23 



28.24 



28.25 



28.20 J L,{t)dt = L„ix) - -^^^^Y 






II 



2! 



I a;Pe-^L„(x) da; := -^ 

''o [(-l)«(m! 



(-!)««! 



if p < n 
)2 if p = n 






(n\)^{x~y) 






L„(a;) = r «"«*-« Jo (2-/^) 



d«. 



lAGUERRE'S ASSOCIATED DIFFERENTIAL EQUATION 



29.1 



xy" + (m + 1 — x)y' + {n — m)y = 



iSOCIATED LAGUERRE POlYN< 



Solutions of 29.1 for nonnegative integers m and n are given by the associated Laguerre polynomials 



29.2 






where Ln(a;) are Laguerre polynomials [see page 153], 



29.3 
29.4 



Lnix) = Ln(x) 

L^{x) = 9 if m > n 



ISSOCIATED LAGUERRE POLYNOMIAI. 



29.S 


l\{x) = -1 




29.10 


Llix) - -6 




29.6 


Llix) = 2x-4 




29.11 


Llix) = 4a;3 - 48a:2 + UAx - 


-96 


29.7 


Llix) = 2 




29.12 


lI(x) = 12«2 - 96a; + 144 




29.8 


Llix) = -3a:2 + 18a: - 


-18 


29.13 


Llix) = 24a; -96 




29.9 


Llix) = ~Gx + 18 




29.14 


Ltix) = 24 





GENERATING FUNCTION FOR L^j 



29.15 



(1 -«)" + ! 



- Lnix) 

2 -r^*** 



155 



156 



ASSOCIATED LAGUERRE POLYNOMIALS 




ORMULi 



29.16 



r-i — Ln + i(x) + {x + m — 2ti—l)Lnix) + n^Lrt-iix) = 

71 -j- 1 



29.17 



■^{Ln{x)) = Ln (X) 



29.18 



■^{x-^e-'^ Ln{x)) = (m~n-l)a:'"-'c-^Ln \x) 



29.19 



x^{Ln{x)} - ix — m)Ln(x) + {m — n — l)Ln (x) 




ORTHOGONALITY 



29.20 



{c««-* Ln(x) Lpix) dx = p¥^n 

A 



29.21 



X 



^ x-e-HLn(xWdx = ^^ 



ORTHOGONAL SERIES 



29.22 

where 
29.23 



f(x) = A^Lm(x) + A^+iLm-nix) + A^ + 2^m+2{x) + 



A^ = 



{k~ 



{k\) 






x^e~^Lh (x) fix) dx 



^SPECIAL RESULTS 



29.24 



-m, . ,, m! nln — m} , n(n ~ l){n ~ m){n ■- m — 1) 

^nix) = (-l)«^^-^|x«-".- -L_^a:«-'«-l + -^ '.^^-^ L-^n-^-- 



, 3;n— m — 2 ^ 



} 



29.25 



r" «. + , rrr^'/M^j (2«-Tn + l)(K!)3 



CHEBYSHEV'S DIFFERENTIAL EQUATION 



30.1 



(l-x2)y" ~ xy' + n'^y = n = 0,l,2,... 




CHEBYSHEV POLYNOMIALS OF TH£ FIRST KIND 




Solutions of 30.1 are given by 
30.2 r„(x) = cos (71 cos-i a!) = a:« 



- C"^x"-2a-a:2) + /'"'\a;"-» (1-052)2 _ 



SPECIAL CHEBYSHEV POLYNOMIALS OF THE FIRST KINI 

8x* - 8fl;2 + 1 

16ar5 - 20x3 + 6a; 

32x« - A8x* + 18x2 _ 1 

64a:' - 112z^ + 56x3 _ ij^ 



30.3 


To(x) = 1 


30.7 


T^ix) 


30.4 


T,{x) = X 


30.8 


Ts(x) 


30.5 


Tzix) = 2a;2-l 


30.9 


Tsix) 


30.6 


Tsix) = 4x^-Sx 


30.10 


Tj{x) 



GENERATING FUNCTION FOR Tn{x) 



30.11 



I- tx 



1-Ztx+ t^ 



2 r„(x)t« 

n=0 



SPECIAL VALUE! 



30.12 T^{-x) = (-l)T„(a;) 

30.13 ^^{l) = 1 



30.14 r„C-l) = (-1)" 

30.15 r2^(0) = (-1)« 



30.16 r2,+,(0) - 



157 



158 



CHEBYSHEV POLYNOMIALS 



Fira^^rFORMuraFo 



30.17 



T„^i{x) - 2xr„(x) + r„_i(a:) = 



30.18 



30.19 



/ 



I T^{x)T„ix) 



J 



-1 Vl-3:2 

1 {r„(x)}2 



da; = m ¥= n 



dx = 



TT if n = 
jr/2 if n= 1,2, . 




30.20 

where 
30.21 



fix) = \A^T^{x) + A,T^(x) ■\- A^T^ix) + 
2 r fix)T^{x} 



-IS 



-1 \/l-x2 



dz 



30.22 



IHEBYSHEV POIYNOMIALS OF TJ 



;OND KIND 



Un(x) = 



sin {(n + 1) cos~^ a:} 
sin (cos""^ a;) 



n+ 1 



a;" — 



n + 1 



g xn-2Cl-x2) + , 5 



m + 1 



.11-4/1 —^212 _ 



x"-4(l-a:2) 



SPECIAL CHEBYSHEV POLYNOMIALS OF THE SECOND KIND 



30.23 


Vcix) = 1 




30.27 


Uiix) '- 


= IQx* - 12a;2 + i 


30.24 


Uiix) ^ 2z 




30.28 


U^x) -- 


= 32x5 _ 32a;3 + 6x 


30.25 


Uzix) = 4a;3- 


- 1 


30.29 


U^ix) = 


= 64x8 _ 80x4 + 24x2 _ i 


30.26 


U^ix) - 8x3- 


4x 


30.30 


Ujix) -- 


= 128x7 _ i92a;5 + gOx^ - 8x 



FUNCTIOI 



30.31 



1 - 2tx + t2 



2 u^{^)tn 

n=0 



CHEBYSHEV POLYNOMIALS 



169 



30.32 U„(-x) = (-l)«t/„(x) 

30.33 UJl) = n+1 



30.34 I7J-1) = (-l)«(n+l) 

30.35 U2n(0) = (-1)« 



30.36 [72,,, (0) = 



30.37 



RECURSION FORMULA FOR Un{x) 



U^^^ix) - 2xU„(x) + t7„_i(x) = 



ORTHOGONALITY 



30.38 



30.39 



30.40 

where 
30.41 



30.42 
30.43 

30.44 
30.45 



30.46 



f Vl-a;2 c;^{iB) C7„(a;) dz = m 9^ n 

j Vr^{U^ix)}^dx = I 

ORTHOGONAL SERIES 

fix) = A^Uoix) + A^U.ix) + A^U^ix) + 



A^ - - f' Vl^^fix) U^ix) 



dx 



RELATIONSHIPS BETWEEN TJx) AND C^J*! 



n(x) = U^{x) - xU„^i{x) 
{l-x^)U^.,{x) = xT^ix) - T^^Ax) 

1 r' r^+iMdij 



t^™{a:) 



_ 1 p ^r. 



a:)Vl--y2 



r„(x) 



1 W VT-^2C7„_,(^) 



= -J 



dv 



GENERAL SOLUTION OF CHEBYSHEV'S DIFFERENTIAL EQUATION 



V = 



AT„ix) + By/T^^^U n-iix) il n= 1,2,3, ... 



A + B sin-i a 



if n= 




31.1 



HYPERGEOMETRIC DIFFERENTIAL EQUATION 

x(l-x)y" + {c-(a+b + l)x)y' - aby = 



Wl 



RGEOMETRIC FUNCTIONS 



A solution of 31.1 is given by 

310 mn h-.-^\ _ 1 , a.6 ^ a(a + l)&(6 + l) a(a+ l)(a + 2)6(6 + 1)(6 + 2) . ^ 

31.2 Fia,b,c,x) - l+-^x+ i.2.c{« + l) ^'+ 1.2-3. c(c+l)(c-F2) *' + 

If a, 6, c are real, then the series converges for — 1 < a; < 1 provided that c — (a + 6) > —1. 



31.3 F(-p,l;l;-a:) = (! + «)" 

31.4 F{l,l;2;-x) = [ln(l + JB)]/a! 

31.5 lim F{l,n;l;x/n) = e* 

31.6 F(^,-^;^;sin2a;) = cosx 

31.7 F(^,l;l;sin2a;) = secx 



SPECIAL CASE! 



31.8 



31.9 F{^,l;l;-x^) = (tan-iic)/a: 

31.10 F{l,p;p;x) = 1/(1 - x) 

31.11 F(n + l,-n;l;il-x)/2) = P„(a;) 

31.12 F(n. -7i; ^; (1 - a:)/2) = r„(x) 



GENERAL SOLUTION OF THE HYPERGEOMETRIC EQUATION 

If c, a — 6 and c — a — b are all nonintegers, the general solution valid for ]«! < 1 is 
31.13 y = A F{a, b; c; x) + Bx^~<' F(a- c + 1, b - c + l;2- c; x) 



31.14 
31.15 
31.16 
31.17 



MISCELLANEOUS PROPERTIES 



F(o, b; c; 1) = 



r(o) r(c - g - 6) 

T(c - a) r(c - 6) 



—Fia, &; c;x) = — F(a + 1, 6 + 1; c + 1; x) 
dx c ^ 

F{a,b:c;x) = r(6)r(c-&) J" ^"'Ml -^)'-'''-'(l -ux)"" du 



F(a, 6; c; k) = (1 - a)c-a-bF(c - a, c - 6; c; x) 



160 




DEFINrriON o 



^B^ApfAC^ffW^^^oP^fff 



32.1 



^{Fim = J €-'tF{t)dt = /(8) 



In general /(a) will exist for s > a where a is some constant. .C is called the Laplace traneform 
operator. 



DEFINITION OF THE INVERSE LAPLACE TRANSFORM OF f{s) 



If ^{F(e)> = /(s), then we aay that F(i) = ^-^{/(s)} is the inverse Laplace transform of /(s). 
Jl~^ is called the inverse Laplace transform operator. 



COMPLEX INVERSION FORMULi 



The inverse Laplace transform of /(s) can be found directly by methods of complex variable theory. 
The result is 



32^ 



F{t) 






c + fT 



where c is chosen so that all the singular points of /(«) lie to the left of the line Re {s} = c in the complex 

s plane. 



161 



162 



LAPLACE TRANSFORMS 



tE OF GENERAL PROPERTIES OF LAPLACE TRANSFORM 






/(«) 


F<t) 


32.3 


afAB) + bU(8) 


oF,(() + &F2(() 


32.4 


fis/a) 


a F(at) 


32.5 


f{B-a) 


e«tF(t) 


32.6 


e-'»m 


(Fit -a) t>a 
[0 t<a 


32.7 


Bf(8)-F(0) 


F'it) 


32.8 


82/(«)-«F(0)-F'(0) 


F"(t) 


32.9 


8»f{8) - s"-»F(0) ~ stt-2F'(0) - • • • - F<«~i> (0) 


FCn)(() 


32.10 


/*(•) 


-tF(t) 


32.11 


/"(8) 


t^F{t) 


32.12 


/(n)(g) 


(_l)ntn p(t) 


32.13 


* 


f F(u)du 

*^0 


32.14 


a* 


J].../^FM.«.=j;<;_%, FM.„ 


32.15 


m 9(8) 


r F{u)G{t~u)du 
•^0 



LAPLACE TRANSFORMS 



163 





/(s) 


Fit) 


32.16 


j m)du 


Fit) 
t 


32.17 


^ r^ 


Fit) = Fit + T) 


1 _ e-sTj " r-iujau 


32.18 


8 




32.19 


-fa/8) 

8 


1 Jo(2-\/wt)F(M)(£M 


32.20 


prn/a/^) 


pin r M-n/2J„(2\/ut)F(M)dM 
•^0 


32.21 


f(s + 1/s) 
s^ + 1 


r' 


1 Jai2\/uit-u))Fiu)du 

•^0 


32.22 


2v^^o 


F(t2) 


32.23 


/(Ins) 
« In s 


f" t^Fiu) , 
J, Tiu+l)^'' 


32.34 


P(s) 

Q(s) 
p(8) = polynomial of degree less than n, 

Q(8) = {8- ai)(8 - aa) ■ ■ • (S - an) 

where ai, aa. • ■ • , "n ^^^ all distinct. 


4 P(«fe) ,«.t 



164 



LAPLACE TRANSFORMS 




IB or 




TSPITTCE TRANSFORMS 





m 


F(i) 


32.25 


1 

8 


1 


32.26 


1 

S2 


£ 


32.27 


^ n-1,2,3.... 


^ , 0! = 1 

{n-1)!' 


32.28 


^ «>o 


r(7i) 


32.29 


1 

8 — O 


e« 


32.30 




fn— 1 aa( 

, 0' = 1 
(n-l)!' 


(8_„)n " l,-.3.... 


32.31 




(n-1 gat 

Tin) 


(a — a)" 


32.32 


1 
82 + a2 


sinaf 
a 


32.33 


8 


cos at 


82 + a2 


32.34 


1 


e^t sin at 
a 


(8 - 6)2 + a2 


32.35 


8- & 


e^"' cos at 


(8 - 6)2 + a2 


32.36 


1 


sinh at 
a 


82 -a2 


32.37 


8 


cosh a£ 


82 - a2 


32.38 


1 


«''* sinh at 
a 


(8 - 6)2 - a2 



LAPLACE TRANSFORMS 



165 





m 


F(i) 


32.39 


8- b 


e"" cosh at 


(8 - 6)2 - o2 


32.40 


1 . , 


gbt — ^t 
b-a 


(s-a)(s-6) "^'' 


32.41 


« . J 


6 — a 


(8-a)is~b) ''^'^ 


32.42 


1 


sin at — at cos at 
2a3 


(s2 + a2)2 


32.43 


8 


£ sin at 
2a 


(S2 + a2)2 


32.44 


82 


sin at + a( cos at 
2a 


(s2 + a2)2 


32.45 


83 


cos o( — ^at sin at 


(82 + a2)2 


32.46 


s2-a2 
(s2 + a2)2 


t cos at 


32.47 


1 


at cosh ai — sinh at 
2a3 


(S2 - o2)2 


32.48 


« 


£ sinh at 
2a 


(a2 - a2)2 


32.49 


«2 


sinh a( + at cosh a( 
2a 


(82 _ a2)2 


32.50 


83 


cosh at + -ia( sinh at ' 


(s2-a2)2 


32.51 


82+ a2 

(82 - a2)a 


t cosh ai 


32.52 


1 


(3 - aH^) sin at - Sat cos at 
8a5 


(«2 + (i2)3 


32.53 


8 


( sin at — at^ cos at 
8a3 


(82 + o2)« 


32.54 


S2 


(1 + a2(2) sin at — at cos at 


(82 + o2)8 


8a3 


32.55 


a3 


3t sin at + at2cosot 
8a 


(a2 + a2)3 



166 



LAPLACE TRANSFORMS 





m 


f(*) 


32.56 


s* 


(3 — a^t^) sin ai + Qat cos o( 
8a 


(s2 + a2)3 


32.57 


S5 


(8 — a^t^) cos a* — lat sin at 
8 


(s2 + a2)3 


32.58 


3s2 - o2 

(s2 + a2)3 


(2 sin at 
2a 


32.59 


83 - 3a^s 

(s2 + a2)3 


^(2 cos at 


32.60 


(g2 + a2)4 


^t^ cos at 


32.61 


g3 _ fl,2g 


t3 sin at 
24a 


(g2 + 0,2)4 


32.62 


1 


(3 + a2t2) sinh at — Sat cosh at 

8a& 


(«2 - a2)3 


32.63 


8 


at2 cosh at ~ t sinh at 
8a3 


(s2 - a2)3 


32.64 


S2 


at cosh at + (a2t2— 1) sinh at 
8a3 


(g2 _ (i2)3 


32.65 


S3 


3f sinh at + at^ cosh at 
8a 


(82 - a2)3 


32.66 


81 


(3 + a2t2) sinh at + 5at cosh at 
8a 


(82 _ 02)3 


32.67 


85 


(8 + a2t2) cosh at + 7at sinh at 
8 


(8= - o2)3 


32.68 


3s2 + a2 

(s2 - a2)3 


t2 sinh at 
2a 


32.69 


83 4- 3a28 
(82 - a2)3 


^(2 cosh at 


32.70 


«4 + 6a2s2 + a-* 

(s2 - a2)4 


■Jt3 cosh at 


32.71 


s3 + a^s 
(82 - a2)4 


t3 sinh at 
24a 


32.72 


1 


gat/z f /- . VSat v/Sat „ ,J 
l-^|V3s.n^2 cos ^2 + e-3.t/2| 


s3 + a3 



LAPLACE TRANSFORMS 



167 





/(s) 


F{t) 


32.73 


s 


eat/2 J VSat , r- . VSat .,J 
3a V°^ 2 + ^ ''" 2 ' 1 


s3 + a3 


32.74 


S2 


^3(.- + i:c"t/acos^/*) 


s3 + a^ 


32.75 


1 


-:r{e3»--cos^,--V3.ni-} 


8^ — a^ 


32.76 


s 


i_^{V3.n4--cos4-.e3..} 


gS — (j3 


32.77 


82 


1 / \/3 o A 

g ( e«t + 2«-''t/2 cos ^^-^—j 


s3 — a3 


32.78 


1 

s4 + 4a4 


^-g (sin at cosh at — cos a* sinh at) 


32.79 


s 


sin a£ sinh at 
2a2 


s4 + 4o4 


32.80 


S2 


^ (sin at cosh at + cos at sinh at) 


«4 + 4a* 


32.81 


s3 


cos at cosh at 


s* + 4a4 


32.82 


1 


^-g (sinh at — sin at) 


s*-a4 


32.83 


8 


. ■^-2(cosh at — cos at) 


s4-a4 


32.84 


S2 


g- (sinh at + sin at) 


s4-a4 


32.85 


S3 


■|-(cosh ot + cos at) 


s4- a* 


32.86 


1 


g-bt _ g-at 


Vs + o + V* + 6 


2(6 - o) V^«^ 


32.87 


1 


erf V^ 




sVs + a 


32.88 


1 


e"* erf Vot 
\/a 


Vs (a — a) 


32.89 


1 


gat j J^ 6 e**** erfc (6V^) I 


y/8 — a+ b 



168 



LAPLACE TRANSFORMS 





f(s) 


i^(0 


32.90 


1 


Jo (at) 


Va^ + a2 


32.91 


1 


/o(at) 


Vs2 - a2 


32.92 




a"/„(aO 


71 > —1 

Vs2 + 0.2 


32.93 




a^IJat) 


\/s2 - a2 


32.94 






gb(»- Vs^ + o') 


Joia^/t{t + 2b)) 


■\/s2 + 02 


32.95 


g-W»»+a» 




iJoia^/t^-b^) t>b 
\o t<b 


Vs2 + a2 


32.96 


1 


tJiiat) 
a 


(s2 + a2)S/2 


32.97 


s 


tJoiat) 


(S2 + a2)3/2 


32.98 


82 


J(i(at) — at J I (at) 


(S2 + o2)3/2 


32.99 


1 


tJiiat) 
a 


(S2 - a2)3.'2 


32.100 


s 


tloiat) 


(S2 - a2)3/2 


32.101 


S2 


/o(at) + atl^iat) 


(S2 ~ a2)3/2 


32.102 


1 e-* 


F{t) = n, n ^ t < n + 1, n = 0, 1, 2, . . . 


8{e»-l) s(l-e-») 
See also entry 32.165. 


32.103 


1 e-s 


[ti 
F{t) = ^r^ 

k = l 
where \t] = greatest integer ^ ( 


8{es — r) 8(1 -re-") 


32.104 


8(e» — r) 8(1— re-") 
See also entry 32.167. 


F(t) = r", n^t<n+l, m = 0, 1, 2, . . . 


32.105 


g-a/s 


cos 2^/at 



LAPLACE TRANSFORMS 



169 





/(«) 


Fit) 


32.106 


g-Q/S 


B\Ti2y/ai 
yfva 


s3/2 


32.107 


e-o/s 


/ \n/2 


gn + l 


32.108 


e-aVs 


V7t 


Vb 


32.109 


e-aVa 


o- • 


2\/n-*3 


32.110 


s 


erf ia/2-/i ) 


32.111 


g-av's 


erfc (a/2V^) 


8 


32.112 


fi-av7 


eb(bt+a) erfc f b\fi + -^\ 
\ 2VV 


Vs (Vs + 6) 


32.113 


g-a/l^ 




gn+l 


32.114 


>»(m) 


g-bt _ e-at 


« 


32.115 


In [(s2 + a2)/a2] 

28 


Ci(ae) 


32.116 


In [{b + a)/a.] 

8 


Si(at) 


32.117 


(Y + In 8] 
s 

y = Euler's constant = .5772156... 


Inf 


32.118 


in(s2 + b2y 


2 (cos at — cos bt) 
t 


32.119 


^2 (y + lns)2 

68 a 
y = Euler's constant = .5772156... 


In^t 


32.120 


In 8 

8 


- (In e + 7) 
y = Euler's constant = .5772156... 


32.121 


InZg 

8 


(In t + y}2 - ^JT^ 
y = Euler's constant = .5772156... 



170 



LAPLACE TRANSFORMS 





m 


F(0 


32.122 


r'(n + 1) - r(m + 1) In « 


t" Int 


32.123 


tan~^ (a/s) 


sin at 


32.124 


tan~' (a/s) 
s 


St (at) 


32.125 


gO/S . 

erfc {yah ) 


g-2V^t 


v^ 


32.126 


/''*''' erfc (s/2o) 


2a „v 


32.127 


/"*"' erfc (s/2a) 


erf (at) 


32.128 


e"* erfc ^fas 


1 




Vir(f + a) 


32.129 


e*" Ei(as) 


1 

t + a 


32.130 


1 
a 


cos 08 -J^— Si(as) l — sin OS Ci{a^) 


1 


t2 + a2 


32.131 


ainoB <-— Si (as) y + cos as Ci (a«) 


t 


t2+ a2 


32.132 




cos as -J ^ — St (as) ^ — sin oa Ci {ax) 


tan -3 {t/a) 


8 


32.133 




sin as ■< ^ — Si (as) > 4- cos as Ci (as) 


1, /t2 + aA 
2'\ a2 ; 


s 


32.134 


r "1^ 


1 , /t2 + a2\ 

-t'\ a2 ; 


32.135 





'^(0 = null function 


32.136 


1 


8(t) = delta function 


32.137 


e-as 


S(f-a) 


32.138 


S 

See also entry 32.163. 


Vit - a) 



LAPLACE TRANSFORMS 



171 





/(») 


^(«) 


32.139 


sinh sa: 
8 sinh sa 


a; . 2 ^ (—1)" . mrx nirt 

— 1- - i -^ — '— sin cos 

a IT „ = 1 n a a 


32.140 


sinh 8x 
8 cosh sa 


4 ^ (-1)« . (2n-l)7ra; . (2n-lVt 

- 2 o — H-sin^ r— ^ — sin^ — '- — 

n- „= J 2n— 1 2a 2a 


32.141 


cosh sx 
8 sinh as 


( , 2 S (-1)" nvx . nirt 
— 1 — > cos sin 

a B- n"^! n a a 


32.142 


cosh sx 
8 cosh sa 


. , 4 ^ (-1)" (2n-lVa! (2n-lVt 

1 H 2 TT -^ COS ^^ r — COS ^ ■= 

T „ = i 2n— 1 2a 2a 


32.143 


sinh sx 
8^ sinh sa 


at , 2a ^ (—1)" . rtirx . nwt 

1 5-2 ^ — 5^sin sin 

a TT^ „=i n'' a a 


32.144 


sinh sx 


, 8a ^ (-1)" . (2n-lVx (2n-lVt 

X ^ ^ 2t 7S 7T9 Sin ^ cos x- — — 

57^ „=i {2n — 1)2 2a 2a 


s2 cosh sa 


32.145 


cosh 8X 
8^ sinh sa 


e2 , 2a ^ (-1)" nirx ( , niTi\ 
„ + 9 2 9 cos 1 cos 1 
2a TT^ „=( n2 a \ a / 


32.146 


cosh sx 
s^ cosh sa 


. , 8a ^ (-1)" (2m -IM . <2n - IM 
^+^2^(2UV°^ 2a «•" 2a 


32.147 


cosh sa; 
s3 cosh sa 


W + x2 a2) ^^"^^ i <-l^" cos^^^^-^^'^^os^^""^''^* 


32.148 


sinh xVT 
sinh aVT 


^ 2 ( l)««e--*'^''-'8in'^"-=' 

a n=l a 


32.149 


cosh x-\f8 
cosh oV? 


h 2 {-l)«-M2n-l)e-<2n-i>Vt/4a'cos^^^^^^^ 
a' „=i ^a 


32.150 


sinh x\fB 
yfs cosh aVi 


- 2 (-l)n-l«-(2«-l>Vt/4a'sin^^^^^^ 
a „-i ^a 


32.151 


cosh x^f8 
yfa sinh aVs 


1+2 1 (_ij„^-nVWe08^^ 

a a „=i a 


32.152 


sinhxVfi^ 
s sinh aVs 


a a- „= 1 n a 


32.153 


cosh a:Vfi 
8 cosh aVs 


1 + ^2 <-^'>-c2«-nVt/4««,o,(2r^-lV« 
a- ^■r'j 2?i — 1 2a 


32.154 


sinh xVs^ 
s2 sinh oVs 


xt , 2a2 " (-1)",, „*^t/„»v ■ «'^a; 


32.155 


cosh a:V* 
s2 cosh aVs 


4(X2 a^\ + t 1^"' -5 ^-^*" «-(2n-l)Vt/4a« cos'^"'^^"'' 



172 



LAPLACE TRANSFORMS 





fie) 


F{t) 


32.156 


JgiixVs) 
8 Jq (iay/T) 


where X^, X2» ■ ■ • ^'^ *^® positive roots of Jq{\) = 


32.157 




n=l Xn-ZitM 

where X^, \2i • ■ ■ are the positive roots of ^o(^) ~ ^ 






Triangular wave function 








F{t) 


32.158 


a«2*^"^(2) 


1- 


/\/\/\/ 






2a 4a 6a 






Fig. 32-1 






Square wave function 


32.159 


itanh(f) 


1- 


F(t) 




la i 2a 1 3a i 4a ' 5a 








1 I ! I L^ 








Fig. 32-2 






Rectified sine wave function 








Fm 


32.160 


e2«2 + ^coth(^2J 


1- 


^\/^\/^\/ 






a 2a 3a ' 






Fig. 32-3 






Half rectified sine wave function 








Fit) 


32.161 


vd 


1- 


r\ r\ / 


(a282 + ^)(l-e-af) 




a 2a Sa 4a 






Fig. 32-4 






Saw tooth wave function 








FW 


32.162 


1 e-<^ 


1- 


^^\^^\^ 


aa^ 8(1-6-08) 




a 2a 3a 4a 






Fig. 32-5 



LAPLACE TRANSFORMS 



173 



32.163 



32.164 



32.165 



32.166 



32.167 



32.168 



f(s) 



s 
See also entry 32.138. 



e-os (! — «-") 



s{l - 6-"^) 
See also entry 32.102. 



e(l - e-*)'' 



1- e- 



8(1 — re-«) 
See also entry 32.104. 



a2s2 + 7r2 



Fit) 



Heavislde's unit function V{t — a) 

F{t) 



1- 



Flg. 32-6 



Pulse function 
Fit) 

1- 



Fig. 32-7 



Step function 

3- 
2- 
1 



2a 



— I— 
3a 



Fig. 32-8 



Fit) =n2, 71 S t<n+l, 71 = 0, 1,2, 



Fig. 32-9 



F(() = r«, n ^ ( < n + 1, n = 0, 1, 2, 



2 3 

Fig. 32-10 



Fit) = 



sin {Trt/a) ^ t g a 




t > a 



Fig. 32-11 



— I— 
4a 





F(t) 


1 

1 






3- 






2- 
1- 






1 \ ^ ^ 



T ' 




33.1 

where 

33.2 



FOI 



^m 



W-iSR^ 



f{^) = I MW cos aX + B{a) sin ax) da 

If" 
A (a) — ~ I fix) COS ax dx 

1 C" 
B{a) = — I f{x) sin ax dx 



Sufficient conditions under which this theorem holds are: 

(i) f{x) and f'(x) are piecewise continuous in every finite interval —L < x < L; 

(ii) 1 |/(a;)| dx converges; 

(iii) f(x) is replaced by ^{f{x + 0) + f{x — 0)} if a; is a point of discontinuity. 



33.3 



33.4 



EQUIVALENT FORMS OF FOURIER'S INTEGRAL THEOREM 



fix 



) = 2~ J j ^^^^ *^°^ aix-u) 

■^rth^; .A *'*! — M^ 



du da 



a= —30 "11= — 



fix) = ^ C e'«* da ( fill) e-*«" du 



du da 



— 00 — 1 



33.5 /(a;) = — I sin ax da I /(w) sin au du 
where fix) is an odd function {f{—x) = —/(a:)]. 

33.6 /(a;) = — ( cos aa; da | /(m) cos au du 
where fix) is an even /uncfcion [f(~x) =f(x)]. 

174 



FOURIER TRANSFORMS 



175 




The Fourier transform of f(x) is defined as 

33.7 ?"{/(«)> = ^(«) = J /(a;)e-*«*dx 
Then from 33.7 the inverse Fourier transform of F(a) is 

33.8 f-HFia)) = f{^) = ^J ^(«)e*«*d« 
We call f{x) and F{a) Fourier transform pairs. 



CONVOLUTION THEOREM FOR FOURIER TRANSFORMS 



If F(a) = Tifix)) and G(a) = Tiffi^)}, then 
33.9 ^ f F{a) G{a) e^=^ da = J fiu)g(x-u) 

where f*g is called the convolution of / and g. Thus 

33.10 7{f*ff} = nnriff} 



du = f*g 




EVAL'S IDENTITY 




If Fia) = nfix)}, then 

33.11 J" \fix)\^dx = ^y \F{a)\^da 

More generally if F(a) = f{f(x)} and G{a) = ^{^(a!)}, then 

33.12 C fix)^)dx = ^ r F{a)GW)da 
where the bar denotes complex conjugate. 




FOURIER SINE TRANSFORMS 

The Fourier sine transform of f{x) is defined as 

33.13 Fsia) = Tsifix)) = r f{x)amaxdx 

Then from 33.13 the inverse Fourier sine transform of Fsia) is 

2 r°° 

33.14 fix) = frM-PsW} = -\ Fsia)Binaxda 




176 



FOURIER TRANSFORMS 




dx 



The Fourier cosine transform of fix) is defined as 

33.15 Fc(a) = TciH^c)) = f f{x)cosax 

Then from 33.15 the inverse Fourier cosine transform of Fc(a) is 

33.16 fix) = f-HFcM) = Ij Fc(«)cosaxd« 




TRANSFORM PAI 






fix) 


J^(«) 


33.17 


jl \x\ < b 
\o \x\ > b 


2 sin 6a 

a 


33.18 


1 


b 


X^+ 62 


33.19 


X 


--g-bot 

o 


X^+ 62 


33.20 


/(n)(a:) 


t"a"F(a) 


33.21 


x^fix) 




33.22 


fibx)e^ 


iK-r) 



FOURIER TRANSFORMS 



177 



SPECIAl FOURIiR SINE TRANSFORMS 



33.23 



33.24 



33.25 



33.26 



33.27 



33.28 



33.29 



33.30 



33.31 



33.32 



33.33 



33.34 



33.35 



33.36 



/(x) 



1 < a: < 6 
a; > 6 



»-» 



a:2 + 62 



,-bi 



a;"—) g—bx 



,-bI* 



--1/2 



a;-" 



Bin bx 



Bin bx 



cos bx 



tan-i (rc/b) 



csc&z 



e2x_i 



fcM 



1 — cos 6a 



He-ba 
2 



a2+ 62 



r(n) sin (n tan-W6) 

(a2 + 62)n/2 



463/2 "« 



/4b 



v^ 



ya"~l CSC (ntr/2) 
2r(7i) 



< m < 2 



1 , fa + b 

2^n;r^ 



B^a/2 a < 6 
3r6/2 a> b 



' a < 6 
7r/4 a=b 
r-/2 a > 6 



2^*"''" 



26**"*^ 26 



f-^K?)~^ 



178 



FOURIER TRANSFORMS 



SPECIAL FOURIER COSINE TRANSFORMS 





/(*) 


Fcia) 


33.37 


Jl < X < & 
to a; > 6 


sin 6a 
a 


33.38 


1 


26 


a2 + (,2 


33.39 


g~bx 


b 


a2 + 62 


33.40 


a;n-l 0-bx 


r{»i) cos (n tan^i a/6) 

(«2 + 62)n/2 


33.41 


e-bx' 


2\6 


33.42 


a: -1/2 


v^ 


33.43 


a;-n 


^«'-isec(W2) o<«<l 
2 r(n) 


33.44 




g-ca _ e-*><^ 


■sra 


33.45 


sin bx 

X 


rw/2 a<b 
• 7r/4 a = 6 
[ a > 6 


33.46 


sin bx^ 


\ IT ( cfi . a^\ 
V86V=°^46-^^"46; 


33.47 


cos bx^ 


V^('°'i^+'^"I^) 


33.48 


sech bx 


^^^^i 


33.49 


cosh (V^ x/2) 
cosh iV^x) 


n? cosh iVTra/2) 
\ 2 cosh (Vw- a) 


33.50 


e-b^ 


■^{cos(26V^) - sin(26v^)> 


V^ 




ELLIPTIC FUNCTIONS 




INCOMPLETE ELirPTIC rNTEGRAL OF THE FIRST KIND 



34.1 



u = F{k 



'*' = r 



d9 



s: 



dv 



where ^ = am u is called the amplitude of u and x = sin 0, and where here and below < fe < 1. 



COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND 



34.2 K = F(ft,W2) = ( 



de 



Vl - *^ sjn2 9 



-i: 



dv 



= fii + (lT*= + (i^T'^ + (i^"^ + 



a 



INCOMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND 



34.3 



Vl - A:2 sin2 tf d9 = I ^ 

n ''o Vl - 



Vl-u2 




ffPW 



THE^SECOND KT 




Vl - fc2 sin2 ^ d# =: I 

^0 



■1 Vl - fc2^2 
Vl-tj2 



dv 



= ii-(iT--feyf-(i^TT 



INCOMPLETE ELLIPTIC INTEGRAL OF THE THIRD KIND 



34.5 






de 



= /: 



dv 



(1 + n sin2 «) Vl - ^^ sin2 « ^^ (1 +mv2) V(l -■u2)(l -ft2i;2) 

179 



180 



ELLIPTIC FUNCTIONS 



34.6 



COMPUTE ELLIPTIC INTEGRAL OF THE THIRD KIND 

da _ r^ dv 

•J n 



Xir/2 



(1 + m 8in2 «) vT-fc^sin^ ^o (1 + nv^) y/(\ - v^){l - k^v^) 




LANDEN'S TRANSFORMATION 



34.7 

This yields 
34.8 



sin 201 



*^"^ = fc + co8 20, *''■ ''^'"'^ " sm[2^,-4>) 



F(fc.0) = f 



d9 



Vl - fc^ 8in2 tf 



2 f*' 
1 + feJ. 



dffi 



\/l-feJsin2(?i 

where k^ — 2\Ic/{l + k). By successive applications, sequences kj, feg. ^3. • - - and <f>i, ^g, 03, . . . are obtained 
such that ft < fci < fcg < feg < ■ ■ - < 1 where lim fe„ = 1. It follows that 



34.9 

where 
34.10 



. 2\/fe , 2V^ ^ ^ 



The result is used in the approximate evaluation of F(ft,0). 




JACOBI'S ELLIPTIC FUNCTIONS 



From 34.1 we deiine the following elliptic functions. 

34. 11 X = sin (am w) = sn m 

34.12 v1^^k2 — cos (am m) = cn« 

34. 1 3 Vl - *=^*^ = Vl -ft2sn2M = dn w 

"We can also define the inverse functions sn-' x, cn~^ x, dn-i x and the following 



34.14 nsM ~ 

34.15 nc« — 

34.16 ndw = 



sn u 

1 
cnw 

1 
dnw 



34.17 



sc w = 



34.18 sd« - ^ 

dnu 



34.19 cd« = ^ 

dnu 



34.20 


CSM — 

snu 


34.21 


. dnM 

dcM = 

cnw 


34.22 


dnu 
dsM — — — ■ 



snM 



34.23 
34.24 

34.25 



sn (m + v) 
en (m + v) 
dn (m + V) 



SUM cni* dntJ + cnM sni) dna 
1 — A;2 sn2 u 8n2 v 

COM en V — snasnudnwdnu 
1 - A;2 sn2 M sn2 11 

dntt dnt? — ft^ snu sni; cna cn-u 
1 — fc2 an2 y gn2 ^ 



ELLIPTIC FUNCTIONS 



181 




34.26 ^sna - cnudnM 



34.27 ^ en u - - sn u dn M 



34.28 -2-dn« 

du 



34,29 ^scu 



— — fc2 sn M en u 



— dc u nc M 




34.30 
34.31 
34.32 



SERIES EXPANSIONS 




cnu = i-|^+(i + 4fc2) j^ - {1 + 4Ak2 + 16k*) fy + • ■ • 

2 A 

dnw = 1 - A:2|y+ fe2(4 + fc2)^_ fc2(i6 + 44fc2 + A^)^+ ... 




34.33 



lf'K,k = If f" "'"^ 



12 32 ^ 52 



= .915965594. 




Let 
34.34 
Then 
34.35 
34.36 
34.37 



PERIODS OF ELLli>TIC FUNCTIONS 



K = 



^0 Vl - fc2 gin2 ' J, 



ff/2 



\/l-fe'2sin2ff 

snM has periods 4^ and 2iii:' 

en u has periods 4K and Sif + 2iK' 

dnu has periods 2K and 4iK' 



where A:' = \1~!^ 



IDENTtTIES INVOLVING EUIPTIC FUNCTIONS 



34.38 sn2tt + cn^w = 1 

34.40 dn2u-A:2cn2« - fe" where )fc' - a/T^^ 



34.42 



34.44 



CH'^M = 



dn 2m + en 2m 



VI 



1 + dn 2u 
— en 2m sn m dn m 



+ cn2u 



cnu 



34.39 dn2u + fc2sn2u = 1 
1 - en 2m 



34.41 



sn-'u — 



1 + dn 2m 



34.43 dn2„ = 1 - fcg + dn 2m + fe' en w 

1 + dn 2m 



34.45 



Vf 



— dn 2u ft sn u en u 



+ dn2M 



dnu 



182 



ELLIPTIC FUNCTIONS 



mmmm 



34.46 snO = 34.47 cnO = 1 34.48 dn = 1 34.49 scO = 34.50 am = 



INTEGRALS 



34.51 I snuiiu = -r In (dn m — /c en u) 

34.52 j en u du = -j- cos~i (dn u) 

34.53 I dnwdw = sin~^snM) 

34.54 I sc u du 



■ , In (dc u + y/T—l^ nc «) 



34.55 I cs u du = In (ns m — ds m) 

34.56 \ cdudu 



-r In {nd M + fc ad u) 



34.57 I dc u du 

34.58 fsdudu 



In (nc M + ac m) 
-1 



In (na m — ca «) 



sin~^ (fe edit) 



34.59 JdaudM 

34.60 I ns M dw = In (ds u — cs «) 



34.61 I ncu 

34.62 I nd u du 



du = 



1 



1 , /, , sew 
In dc M + 



VH^ 



coB~i (cdu) 




LEGENDRE'S RELATION 



34.63 

where 
34.64 

34.65 



EK' + E'K - KK' = v/2 



E 



- r^ 



fc2 8in2 e d$ 



•^ n 



■n/2 




dg 



J-'tt/Z , /•jr/2 fla 

-^0 Vir^fc^i 



Vr^^fc2sin2ff 
sin^ff 



ERROR FUNCTION erf (a;) = — r f e"" 



du 



35.1 erf(x) = ~(x-^+^2, ^.3, 



+ 



35.2 erf («) 



1 - 



1 -7^4 + 



1-3 1*3'5 



+ 



^^\- 2a:2 ' (2a;2)2 (2x2)3 
35.3 erf(-x) = -erf(x), erf (0) ^ 0, erf('») = 1 



COMPLEMENTARY ERROR FUNCTION erfc(a:) = 1 - erf (x) = — = f e "' d« 



35.4 erfc («) 

35.5 erfc (a;) 

35.6 erfc (0) 



2_ / _ a:3 gs _ z? 

V^r 3-1! ^5-2! 7-3! 






+ 



yfi^xV 23:2 ' (2a:2)2 (2x2)S 
1, erfc{<") = 



J"" fi-" 
— - 



du 



35.7 Ei{x) = 

35.8 £i(a^) = 

35.9 Eiix) - 



^i 1 
—7 — In X + I - 

— y — \nx + 



_ »— u 



dw 



+ 



I'l! 2-2! ' 3*3! 



Ill 1-11+2|_|| + 



35.10 EH-^) = 



35.11 Si(x) = 



SINE INTEGRAL 



x3 _^ rS 



Si{x) = f 






({U 



1-1! 3-3! 5*5! 7'7! 



£ _ sing /l _ 3! 5! _ 
2 a; la; arS a:5 



+ 



35.12 Si{x) ~ 

35.13 Si(-x) = -Si(a:), Si(0) = 0, Si(«) = W2 



cos a; / _ 2! 4! _ 

X \^ ^"^^ 



183 



184 MISCELLANEOUS SPECIAL FUNCTIONS 

COSINE INTEGRAL Ci{x) = C'^^^du 

35.14 Ci{x) = -y-lnx+ Ci^l^^^du 

J, 

35.15 Ci(x) = -y - \nx + ^ - ^^ + ^ ^ + ■-■ 

2-2! 4-4! 6-6! 8-8! 

35.16 Ci(x) ~ ^^^(l-^+^ A-Sin^/ 2!,4! \ 

X \x x^^ x^ J X y- x'^^x* J 

35.17 Ci('*) = 

FRESNEL SINE INTEGRAL S{x) = J^ j'Jsinu^ du 

\^V3-1! 7-3! 11-5! 15-7! ^ 

35.19 Six) ~ l-~^Jico.x2)(l~llA+ll3^_ __\ /j^_l^S^ 

2 ^/2^\ V 22a;5 2^x^ J ^ ^"^ " ' ' \2x^ 23ar' 

35.20 S(-x) = -Six), 5(0) = 0, S{«) = -^ 



FRESNEL COSINE INTEGRAL C{x) = .pfcostt^dtt 




35.2, C(.) =^(^-_^,_4_^_-.,...) 



35 



35.23 a-x) - -C(a;), C(0) - 0, C(~) - 1 



RIEMANN ZETA FUNCTION ^a:) = TF + 4 + 4 + 

35.24 «., = ^r"-Hilf_,,, ,>! 



35.25 ni-a;) = 2i-^;r-^r(a:) co3(o-a:/2)j:(a:) [extension to other values] 

-Iy2fc 



22k-V2fc5 
35.26 m) = ~7^J~ k = 1,2.3.... 



TRIANGLE INEQUALITY 



36.1 
36.2 



|ai] - Ittal ^ lai + ojl ^ [aj + [oal 
|ai + fla + ■ . ■ + a„| g ]a,I + [ojl + . - . + |a„| 




CAUCHY-SCHWAR2 INEQUALITY 



36.3 \a,b, + 0362 + ■ . . + a„feJ2 ^ (|a,|2 + la^I^ + • - ■ + \aM\h\^ + I62P + ' • " + i6„|2) 

The equality holds if and only if ai/6, - ag/fca = ■ • • = a„/6„. 



INEQUALITIES INVOLVING ARITHMETIC, GEOMETRIC AND HARMONIC MEANS 

If ^ . G and H are the arithmetic, geometric and harmonic means of the positive numbers a,, a,,..., a, 

H ^ G ^ A 



then 
36.4 

where 

36.5 A = "i + °2 + • • • + '^n 



36.6 G = ^a,a^...a, 36.7 ^ .= 1 f 1 + 1 + . . . + A 



The equality holds if and only if aj = Og = • ■ • = a„. 



H - nVo +a + •■• + 




36.8 

where 
36.9 



HOLDER'S INEQUALITY 

kifri + ^262 + • ■ ■ + a„6„| ^ (|a,|p + la^lP + . . . + Kji>)1/p(|6,|<. + [ft^i^ + . . . + |(,j,ji/. 



- + ^ = 1 P>1. 9>1 



duces 



The equality holds if and only if W-Vil-.l = Wp-./|6,| = • • • = |a,|.-V|M, For p = , = 2 it 



re- 



186 



186 



INEQUALITIES 



CHEBYSHEV'S INEQUALITY 

If tti ^ a2 ^ ' ■ ■ ^ a„ and 61 S 63 = ■ ■ • = &„> then 



n 



n 



n 



36.10 

or 

36.11 {ai + a2+ ■•■ +«„)(&! + &£+ ••■+&„) = n(ai&i + a262+ •■ ■ + a„6„) 



MINKOWSKI'S INEQUALITY 

If Qi, a^, . . ., a„, bi, 6,, . . . b„ are all positive and p > 1, then 

36.12 {(ai+fe,)P+ (02+62)''+ ■■■ + (On+fiJ'*}^^'' ^ (ai + 02+ ■■■+0^^" + ib'l + bl+ ■■■ + 6^)i/p 
The equality holds if and only if O1/61 = 02/62 = ■ ■ - = o.n^f>n- 



CAUCHY-SCHWARZ INEQUALITY FOR INTEGRALS 



I/: 



36.13 II f{x)g{x)dx 

The equality holds if and only if f{x)/g(x) is a constant. 



j''\f{x)\-^dx\Ij^\g{x)\^dx 



HOLDER'S INEQUALITY FOR INTEGRALS 

36.14 J \fix)g{x)\dx ^ if \f{x)\''dxi \j \9{x)\''dx 

where l/p + 1/g = 1, p > 1, g > 1. If p = g - 2, this reduces to 36.13. 
The equality holds if and only if \f{x)\f-^/\g{x)\ is a constant. 



i/n 



MINKOWSKI'S INEQUALITY FOR INTEGRALS 



If P > 1, 
36.15 



<j \nx) + g(x)\^dx^ ^ ij \nx)\^dx^ + U \9(x)\'>dx 



i/p 



The equality holds if and only if f{x)/g{x) is a constant. 




37.1 cot a; = 



X "^ 2a: j^2_^+^2_4^2 + x2_9^ + 



37.2 



CSCX - ^ ^^\^z_^ -2^2 + ^^—^ 



37.3 sec X = Av 



' .+ = 



.2 _ 4a;2 9^.2 _ 4a;2 ^ 2B;r2 - 4a:2 



37.4 tan x = 8a; -I ^ h ^ I- . ^ u 

lan ic »* \2 _ 4a;2 ^ 9^ - 43^2 + 26772 - 4a:2 ^ 



37.5 



sec^a; = 



4 J 1 + 1 + 1 , 1 , 



37.6 csc^ a; = 



-1+ 1 + 1 + 1 I 1 + 

a;2 (ar-;r)2 ^ (a: + ,r)2 ^ (a: - 2,r)2 ^ (a; + 2^)2 ^ 



37.7 coth X = 



1 + 0-.}^L—^ 1 , 1 . 

X U2 + ,r2 ^ ^ + 4^ "^ a:2 + 9^ + 



37.8 csch X = 



l_2x-^-^^ ^ 

X |a;2 + B^ a;2 + 4^ "f' ^2 + 9^ 



1 + ^ 



37.9 sechx = 4^ 



^+ 5 



572 + 4x2 9^2 + 43.2 ^ 25^ + 43.2 



37.10 tanhx = 



^* "^ s-2 -f 4x2 "•■ 9^ + 4a;2 + 25:r2 + 4x2 "^ 



187 



3B.1 



sina: = 



.u_-Vi_- ,_^ 



38.2 



C08X = 



-^Vi-|S)(i-^. 



38.3 sinh x 



^(i + 5)(i + 5)(i+i. 



38.4 cosh X = 



-^)(-is)(^-^; 



See also 16.12, page 102. 



38.6 MX) = 1-^ 1-- 1- 



where Xj, X2, X3, . . . are the positive roots of Jq{x) = 0. 



38.7 JM = .fi-gyi-E!yi_E^ 



where \i, X2, X3> . . . are the positive roots of Ji(x) = 0. 



38.8 



sin X _ 
X 



X X X X 

COS - cos — COS -- cos — - 
2 4 8 16 



38.9 



2 2 4 4 6 6 
1 ' 3* 3' 5'5'7 



This is called Wallis' product. 



188 



BINOMIAL DISTRIBUTION 



39.1 






p>0, g>0, p + q = 1 



POISSON DISTRIBUTION 



39.2 



*(«:)= 2 ^ X > 



HYPERGEOMETRIC DISTRIBUTION 



39.3 



*(«) = 2 



t/\n — t 



t<x /r + s 
n 



NORMAL DISTRIBUTION 



39.4 



X) = ^f e-'Ut 



<i.(x) = 



STUDENT'S t DISTRIBUTION 



39.5 



y/^ ~r(n/2) 



1 V 2 / /•■ 
*^^* = JZZ T(n/2) J 



1+^ 
n 



-(n + I)/2 



d( 



CHI SQUARE DISTRIBUTION 



39.6 



1 r^ 



F DISTRIBUTION 



39.7 



*(«) = 



r(n,/2) r(n2/2) 



r t"./2(n2 + nit)-<"i+»*5/2d( 



189 



40 



SPECIAL MOMENTS 
OF INERTIA 



The table below shows the moments of inertia of various rigfid bodies of mass M. In all cases it is 
assumed the body has uniform [i.e. constant] density. 



TYPE OF RIGID BODY 


MOMENT OF INERTIA 


40.1 Thin rod of length a 




(a) about axis perpendicular to the rod through the center of 
mass, 

(6) about axis perpendicular to the rod through one end. 


40.2 Rectangular parallelepiped with sides a, b, c 


J^M(a2 + (,2) 


(a) about axis parallel to c and through center of face ah, 
{V) about axis through center of face be and parallel to c. 


40.3 Thin rectangular plate with sides a,b 


JLM(a2 + 62) 


(a) about axis perpendicular to the plate through center, 
(6) about axis parallel to side 6 through center. 


40.4 Circular cylinder of radius a and height h 




(a) about axis of cylinder, 

(6) about axis through center of mass and perpendicular to 
cylindrical axis, 

(c) about axis coinciding with diameter at one end. 


._ - Hollow circular cylinder of outer radius a, 
inner radius 6 and height h 


^Af {a2 + 62) 
JLM(3a2 + 362 + fe2) 

r^MiZa^ + 362 + 4^2) 


(a) about axis of cylinder, 

(6) about axis through center of mass and perpendicular to 
cylindrical axis, 

(c) about axis coinciding with diameter at one end. 



190 



SPECIAL MOMENTS OF INERTIA 



191 



40.6 Circular plate of radius a 


iMa2 


(a) about axis perpendicular to plate through center, 
(6) about axis coinciding with a diameter. 


-_ _ Hollow circular plate or ring with outer radius a 
and inner radius 6 


^M{a^ + 62) 
lM(a2 + 62) 


(a) about axis perpendicular to plane of plate through center, 
(6) about axis coinciding with a diameter. 


40.8 Thin circular ring of radius a 


Ma2 
^Ma2 


(a) about axis perpendicular to plane of ring through center, 
(6) about axis coinciding with diameter. 


40.9 Sphere of radius a 


§Mo2 
lMa2 


(a) about axis coinciding with a diameter, 
(6) about axis tangent to the surface. 


40.10 Hollow sphere of outer radius a and inner radius 6 


f M(a5 - 65)/((i3 - 68) 
f M(a5 - 65)/(a3 - 63) + Ma^ 


(a) about axis coinciding with a diameter, 
(6) about axis tangent to the surface. 


40.11 Hollow spherical shell of radius a 


2Ma2 


(a) about axis coinciding with a diameter, 
(6) about axis tangent to the surface. 


40.12 Ellipsoid with semi-axes a,b,c 


|M(6a2 + 62) 


(a) about axis coinciding with semi-axia c, 

(6) about axis tangent to surface, parallel to semi-axis c and 
at distance a from center. 


40. 1 3 Circular cone of radius a and height h 


^M(a2 + 4fe2) 
|M(4a2 + fe2) 


(a) about axis of cone, 

(6) about axis through vertex and perpendicular to axis, 

(c) about axis through center of mass and perpendicular to axis. 


40.14 Torus with outer radius a and inner radius 6 


lM{Qa^ - lOab + 562) 


(a) about axis through center of mass and perpendicular to 
plane of torus, 

(6) about axis through center of mass and in the plane of the 

torus. 




Length 



Area 



Volume 



Speed 

Density 

Force 



Energy 



Power 



Pressure 



1 kilometer (km) 

1 meter (m) 

1 centimeter (cm) 

1 millimeter (mm) 

1 micron (ju) 

1 millimicron (m^u) 

1 angstrom (A) 



— 1000 meters (m) 

— 100 centimeters (cm) 
= 10-2 m 

^ 10-3 m 

— 10-8 m 
= 10-!' m 
= 10-10 m 



1 inch (in.) 
1 foot (ft) 
1 mile (mi) 
1 mil 

1 centimeter 
1 meter 
1 kilometer 



2.540 cm 
30.48 cm 
1.609 km 
10-3 in. 
0.3937 in. 
39.37 in. 
0.6214 mile 



1 square meter (m2) ^ 10.76 ft^ 
1 square foot (f t2) - 929 cm2 



1 square mile (mi2) = 640 acres 
1 acre = 43,560 ft2 



1 liter (/) = 1000 cm^ = 1.057 quart (qt) - 61.02 in3 = 0.03532 ft^ 

1 cubic meter (m^) = 1000 I = 35.32 ft^ 

1 cubic foot (ft3) - 7.481 U.S. gal = 0.02832 m^ - 28.32 I 

1 U.S. gallon (gal) ^ 231 in^ - 3.785 I; 1 British gallon = 1.201 U.S. gallon = 277.4 in* 

1 kilogram (kg) =^ 2.2046 pounds (lb) = 0.06852 slug; lib = 453.6 gm = 0.03108 slug 
1 slug = 32.174 lb = 14.59 kg 

1 km/hr ^ 0.2778 m/sec ^ 0.6214 rai/hr = 0.9113 ft/sec 
1 mi/hr = 1.467 ft/sec = 1.609 km/hr = 0.4470 m/sec 

1 gm/cm3 = 103 kg/m3 = 62.43 Ib/ft^ = 1.940 slug/ft3 
1 lb/ft3 = 0.01602 gm/cm3; 1 slug/ft^ - 0.5154 gm/cm^ 

1 newton (nt) - 10^ dynes = 0.1020 kgwt = 0.2248 Ibwt 

1 pound weight (Ibwt) ~ 4.448 nt = 0.4536 kgwt = 32.17 poundais 

1 kilogram weight (kgwt) ^ 2.205 Ibwt = 9.807 nt 

1 U.S. short ton = 2000 Ibwt; 1 long ton = 2240 Ibwt; 1 metric ton = 2205 Ibwt 

1 joule = Intm = 10^ ergs = 0.7376 ft Ibwt ^ 0.2389 cal = 9.481 X lO^^ Btu 

1 ft Ibwt ^ 1.356 joules ^ 0.3239 cal = 1.285 X 10-3 Btu 

1 calorie (cal) = 4.186 joules = 3.087 ft Ibwt = 3.968 X 10-3 Btu 

1 Btu (British thermal unit) - 778 ft Ibwt = 1055 joules = 0.293 watt hr 

1 kilowatt hour (kw hr) = 3.60 X 10« joules - 860.0 kcal = 3413 Btu 

1 electron volt (ev) = 1.602 X IQ-^^ joule 

1 watt = 1 joule/sec = 10' ergs/sec = 0.2389 cal/sec 

1 horsepower (hp) ^ 550 ft Ibwt/sec = 33,000 ft Ibwt/min = 745.7 watts 

1 kilowatt (kw) = 1.341 hp = 737.6 ft Ibwt/sec = 0.9483 Btu/sec 

1 nt/m^ - 10 dynes/cm2 = 9.869 X 10-8 atmosphere = 2.089 X 10-2 lbwt/ft2 
1 lbwt/in2 - 6895 nt/m^ = 5.171 cm mercury = 27.68 in. water 

1 atmosphere (atm) = 1.013 X 10^ nt/m2 = 1.013 X 10^ dyne3/cm2 = 14.70 lbwt/in2 
= 76 cm mercury = 406.8 in. water 



192 



Part n 



TABLES 



SAMPLE PROBLEMS 

ILLUSTRATING USE OF THE TABLES 



COMMON LOGARITHMS 

1. Find log 2.36. 

We must find the number p such that lO" - 2.36 = A'. Since 10" = 1 and 10^ = 10, p lies 
between and 1 and can be found from the tables of common logarithms on page 202. 

Thus to find log 2.36 we glance down the left column headed N until we come to the first two digits, 
23. Then we proceed right to the column headed 6. We find the entry 3729. Thus log 2.36 = 0.3729, 
ie. 2.36 = 100-3'29. 

2. Find (a) log 23.6, (6) log 236, (c) log 2360. 

From Problem 1, 2.36 = 10''-3729, jhen multiplying successively by 10 we have 
23.6 = 10'-3'29, 236 = 102-3729^ 2360 = 103-3729 

Thus 

(a) log 23.6 = 1.3729 

(b) log 236 = 2.3729 

(c) log 2360 = 3.3729. 

The number .3729 obtained from the table is called the mantissa of the logarithm. The number 
before the decimal point is called the characteristic. Thus in (6) the characteristic is 2. 

The following rule is easily demonstrated. 

Rule 1. For a number greater than 1, the characteristic is one less than the number of digits before 
the decimal point. For example since 2360 has four digits before the decimal point, the 
characteristic is 4 — 1 — 3. 

3. Find (a) log .236, (6) log .0236, (c) log .00236. 

From Problem 1, 2.36 = 10''-3'='29. Then dividing successively by 10 we have 

.236 = 10*>3ra9-J = 1Q9.3729-10 = 10~-8271 

.0236 = 100-3729-2 = X08-3729-10 == 10-16271 

.00236 = 1003729-3 - IO'3729-IO = 10-2-6271 

Then 

(a) log .236 ^ 9.3729 - 10 - -.6271 

(6) log .0236 ^ 8.3729 - 10 ^ -1.6271 

(c) log .00236 ^ 7.3729 - 10 =^ -2.6271. 

The number .3729 is the mantissa of the logarithm. The number apart from the mantissa [for 
example 9 - 10, 8 — 10 or 7 — 10] is the characteristic. 

The following rule is easily demonstrated. 

Rule 2. For a positive number less than 1, the characteristic is negative and numerically one more 
than the number of zeros immediately following the decimal point. For example since .00236 
has two zeros immediately following the decimal point, the characteristic is —3 or 7 — 10. 

194 



SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 195 

4. Verify each of the following logarithms. 

(a) log87.2. Mantissa - .9405. characteristic - 1; then log 87.2 = 1.9405. 

(b) log 395,000 = 5.5966. 

(c) log .0482. Mantissa - .6830, characteristic - 8 - 10; then log.o .0482 - 8.6830 - 10. 
(rf) log .000827 - 6.9175 - 10. 

5. Find log 4.638. 

Mantissa of log 4640 .. .6665 Mantissa of log 4.638 = .6656 + (.8)(.0009) 

Mantissa of log 4630 = .6656 _ (.RRt * * j- -^ 
— .d663 to four digits 

Tabular difference - .0009 Then log 4.638 = 0.6663 

(6656+')!"'' '"' P^°P-*--^ P-ts table on page 202 can be used to give the mantissa directly 

6. Verify each of the following logarithms. 
(a) log 183.2 ^ 2.2630 (2625 + 5) 
(6) log 87,640 - 4.9427 (9425 + 2) 

(c) log .2548 =9.4062-10 (4048 + 14) 

(d) log .009848 = 7.933 - 10 (9930 + 3) 

COMMON ANTILOGARITHMS 

7. Find (a) antilog 1.7530, (b) antilog (7.7530 - 10) 

digits before the decimal point. T.^TeZ^f, .ll^Z'i^ '' '' ^'^^^ ^'^ '^^ 

^'* ac'tlris'ril'V-'lo^th?' '"h' 5662 corresponding to the mantissa .7530. Then since the char- 

Thnrfl I' ""^^"^ """'* ^"^^^ **° ^«^°s immediately following the decimal noint 

Ihus the required number is .005662. ^"'luwing me aecimal point. 

8. Find antilog (9.3842 - 10). 

on page%rwlTa;f' ' "" '^'"^^" ''''' ^"' •'''' »"^ ^ ™-* "^ interpolation. From the table 

Number corresponding to.3850 = 2427 Given mantissa = 3842 

Number corresponding to .3840 = ^42]_ Next smaller mantissa - .3840 

Tabular difference = 6 Difference = "i^^ 

Then 2421 + ^(2427-2421) ^ 2422 to four digits, and the required number is 0.2422. 
The proportional parts table on page 204 can also be used. 

9. Verify each of the following antilogarithm. 

(a) antilog 2.6715 = 469.3 

(b) antilog 9.6089 -10 = .4063 

(c) antilog 4.2023 = 15.930 



196 SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 



COMPUTATIONS USING LOGARITHMS 

j^ ^ ^ (784.6)(.0431) ^ j^gp ^ log 784.6+ log .0431 - log 28.23. 
28.23 

log 784.6 = 2.8947 

(+) log .0431 =^ 8.6345-10 

11.5292-10 

{-) log 28.23 = 1.4507 



logP = 10.0785-10 = .0785. Then P = 1.198. 

Note the exponential significance of the computation, i.e. 

(784.6)(.0431) ^ (102.B347)(10a.e345-10) ^ ^q2.,M7 + S.^Z,B-10-IA,07 = 10.0785 ^ 1198 

28.23 101-4507 

11. P = (5.395;8. logP = 8 log 5.395 = 8(0.7320) - 5.8560, and P ^ 717,800. 

12. P = V38^2 = (387.2)>/2. logP = | log 387.2 = ^(2.5879) = 1.2940 and P = 19.68. 

13. P = ^.08317 = (.08317)1/5. log P - ^ log .08317 = ^(8.9200-10) = f, (48.9200 - 50) - 9.7840-10 
and P = .6081. 

14. P = V-003654 ( 18-3 7P j^^p ^ 4 log .003654 + 3 log 18.37 - (4 log 8.724 + ^ log 743.8) 

(8.724)4 ^743.8 

Numerator N Denominator D 

I log .003654 - ^(7.5628-10) 4 log 8.724 = 4(0.9407) = 3.7628 

= ^(17.5628-20) = 8.7814-10 ^ log 743.6 = 1(2.8714) = 0.7178 

3 log 18.37 ^ 3(1.2641) = 3.7923 Add: log D = 4.4806 

Add: logiV = 12.5737-10 

logN = 12.5737-10 
(-) logZ) - 4.4806 



logP =^ 8.0931-10. Then P = .01239 



NATURAL OR NAPIERIAN LOGARITHMS 

15. Find (a) In 7.236, (6) In 836.2, (c) In .002548. 
(a) Use the table on page 225. 



In 7.240 = 1.97962 
In 7.230 = 1.97824 



Tabular diflference = .00138 
Then In 7.236 ^ 1.97824 4- ^(.00138) = 1.97907 

In terms of exponentials this means that e^-^'^o' - 7.236. 

(6) As in part (a) we find 

In 8.362 = 2.12346 + 3^^(2.12465 - 2.12346) = 2.12370 

Then 

In 836.2 ^ In (8.362 X 102) = log 8.362 + 2 In 10 ^ 2.12370 + 4.60517 = 6.72887 

In terms of exponentials this means that ee-'^ss? = 836.2. 

(c) As in part (a) we find 

In 2.548 - 0.93216 + 3%{0.93609 - 0.93216) = 0.93530 

In .002548 = In (2.548 X lO"-*) = In 2.548 - 3 In 10 = 0.93530-6.90776 - -5.97246 
In terms of exponentials this means that g- 5-97246 - .002548. 



SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES X97 



TRIGONOMETRIC FUNCTIONS (DEGREES AND MINUTES) 

16. Find (a) sin 74°23', (6) cos 35°42', (c) tan 82°56'. 

(a) Refer to the table on page 206. 

sin 74°30' = .9636 

sin 74°20' = .9628 

Tabular difference - .0008 

'^*'*" sin 74<'23' - .9628 + ^(.0008) = .9630 

{&) Refer to the table on page 207. 

cos35''40' - .8124 

cos 35^60' = .8107 

Tabular difference — .0017 

"T^^^" cos 35^42' - .8124 - ^(.0017) - .8121 

" cos 35^42' ^ .8107 + ^(.0017) = .8121 

(c) Refer to the table on page 208. 

tan82°60' = tan 83°0' = 8.1443 
tan82°50' ^ 7.9530 

Tabular difference = .1913 

■^^^^ tan82°56' = 7.9530 + -^^(.1913) = 8.0678 

17. Find (a)cot45°16', (6) sec 73°48', (c) esc 28°33'. 

(a) Refer to the table on page 209. 

cot45°10' = .9942 
cot45°20' := .9884 



Tabular difference = .0058 
^*^^" cot45°16' = .9942 - ^(.0058) = .9907 

°' cot45'^I6' - .9884 + ^(.0058) = .9907 

(6) Refer to the table on page 210. 

sec73°50' - 3.592 

sec73°40' = 3.556 

Tabular difference =: .036 

'^''^" sec 73^48' = 3.556 + ^(.036) = 3.585 

(c) Refer to the table on page 211. 

csc28'=30' - 2.096 

csc28''40' = 2.085 

Tabular difference = ,011 

■"•^ csc28='33' - 2.096 - ^(.011) = 2.093 

°' csc28°33' = 2.085+ ^^(.011) - 2.093 



198 SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 

INVERSE TRIGONOMETRIC FUNCTIONS (DEGREES AND MINUTES) 

18. Find (a) sin-i (-2143), (6) cos-> (.5412), (c) tan-i (1.1536). 

(a) Refer to the table on page 206. 

sin 12^30' = .2164 

sinl2°20' = .2136 



Tabular difference = .0028 

Since .2143 is -^l-^^ ~ -2136 _ j ^^ ^^^ ^ between .2136 and .2164, the required angle is 

.0028 * 

12'*20' + i(lO') = 12°22.5'. 

(6) Refer to the table on page 207. 

cos57°10' ^ .5422 

cos 57=^20' = .5398 



Tabular difference - .0024 
Then cos-M-5412) = 57°20' - i^^^^^^^^dO') - 57^4.2' 

or eos->(.5412) - 57n0' + i^l?^=2^(10'} - 57^4.2' 

(e) Refer to the table on page 208. 

tan49°10' = 1.1571 

tan49°0' ^ 1.1504 



Tabular difference — .0067 
Then tan-M1.1536) = 4900- + 1-1536^- 1.1504 ^^p,^ ^ ^904.8' 

Other inverse trigonometric functions can be obtained similarly. 



TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS (RADIANS) 

19. Find (a) sin (.627), (6) cos (1.056), (c) tan (.153). 

(a) Refer to the table on page 213. 

sin (.630) - .58914 

sin (.620) = .58104 

Tabular difference — .00810 

Then sin (.627) - .58104 + -^^(.00810) = .58671 

(6) Refer to the table on page 214. 

cos (1.050) = .49757 

cos (1.060) = .48887 



Tabular difference = .00870 
Then cos (1.056) = .49757 - ^(.00870) = .49235 

or cos (1.056) = .48887 + ^(.00870) = .49235 

(c) Refer to the table on page 212. 

tan (.160) = .16138 

tan (.150) = .15114 



Tabular difference — .01024 
Then tan (.153) = .15114 + ^(.01024) = .15421 

Similarly other trigonometric functions are obtained. 



SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 199 



20. Find sin"' (.512j in radians. 

Refer to the table on page 213. 



sin (.540) ~ .51414 

sin (.530) = .50553 

Tabular difference = .00861 



.512 - .50553 
.00861 

Similarly the other inverse trigonometric functions are obtained. 



Then sin-M.512) - .530 + iii^--J?^ f.Ol) - .5375 radians 



COMMON LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 

21. Find (a) log sin 63 = 17', (6) log cos 48=" 44'. 

(a) Refer to the table on page 217. 

Iogsin63°20' = 9.9512 - XO 

logsin63°10' = 9.9505-10 

Tabular difference = .0007 

Then log sin 63°17' = 9.9505 - 10 -f -j:^(.0007) = 9.9510 - 10 

(6) Refer to the table on page 219. 

log cos 48°40' = 9.8198 - 10 

logcos48°50' = 9.8184-10 

Tabular difference = .0014 

Then logcos48°44' = 9.8198 - 10 - 3^(.0014) = 9.8192-10 

o^ logcos48°44' ^ 9.8184 - 10 + -j^(.0014) = 9.8192-10 

Similarly we can find logarithms of other trigonometric functions. Note that log sec x = -log cos x, 
log cot ic — -log tan a;, log esc ac — -log sin x. 

22. If log tan a: - 9.6845-10, finds:. 

Refer to the table on page 220. 

Iogtan25°50' = 9.6850-10 

]ogtan25°40' = 9.6817-10 

Tabular difference = .0033 

Then :. = 25°40- + ^-^«^5- 9.6817 ^^Q,^ ^ ^^.^^ g, 

.UUoo 

CONVERSION OF DEGREES, MINUTES AND SECONDS TO RADIANS 

23. Find 75° 28' 47" in radians. 

Refer to the table on page 223. 

70° = 1.221730 radians 

5° = .087267 

20' - .005818 

8' = .002327 

40" = .000194 

7^ - .000034 

Adding, 75° 28' 47" = 1.317370 radians 



200 SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 

CONVERSION OF RADIANS TO DEGREES, MINUTES AND SECONDS 

24. Find 2.5-17 radians in degrees, minutes and seconds. 

Refer to the table on page 222. 

2 radians - 114° 35' 29.6" 

.6 = 28° 38' 52.4" 

.04 = 2° 17' 30.6" 

.007 = 0° 24' 3.9" 

Adding, 2.547 radians = 144° 114' 116.5" - 145° 55' 56.5" 

CONVERSION OF RADIANS TO FRACTIONS OF A DEGREE 

25. Find 1.382 radians in terms of degrees. 

Refer to the table on page 222. 

1 radian = 57.2958° 

.3 = 17.1887° 

.08 = 4.5837° 

.002 = .1146° 



Adding, 1.382 radians = 79.1828° 

EXPONENTIAL AND HYPERBOLIC FUNCTIONS 

2G. Find (a) e->-^'*, (fc) e--i5B. 

(a) Refer to the table on page 226. 

e5.30 ^ 200.34 
e5.2o ^ 181.27 



Tabular difference = 19.07 
Then «5.24 = I81.27 + -^(19.01) = 188.90 

(6) Refer to the table on page 227. 

e--i50 = ,86071 

e-.i6o ^ .85214 



Tabular difference = .00857 
Then e" >S8 = .86071 - ^^(.00857) = .86385 

or e->58 = .85214 + ^(.00857) = .85385 



27. Find (ct) sinh (4.846), (b) sech (.163). 
(a) Refer to the table on page 229. 



sinh (4.850) = 63.866 
sinh (4.840) = 63.231 



Tabular difference ~ .635 
Then sinh (4.846) = 4.840 + ^(.635) = 5.221 

(6) Refer to the table on page 230. 

cosh (.170) = 1.0145 

cosh (.160) = 1.0128 

Tabular diflference = .0017 

Then cosh (.163) = 1.0128 + ^(.0017) = 1.0133 

and so sech (.163) = = — - — = .98687 

^ cosh (.163) 1.0133 



SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 201 

28. Find tanh-' (.71423). 

Refer to the table on page 232. 

tanh (.900) = .71630 

tanh(.890) - .71139 
Tabular difference = .00491 

Then tanh-' (.71423) = .890 + ^HM^^L^Zll^dO) = .8958 

.00491 ' 

INTEREST AND ANNUITIES 

29. A man deposits $2800 in a bank which pays 5% compounded quarterly. What will the deposit 
amount to in 8 years? 

There are n = 8 • 4 = 32 payment periods at interest rate r - .05/4 - .0125 per period. Then the 
amount is 

A = $2800(1 + .0125)32 = $2800(1.4881) = $4166.68 

using the table on page 240. 

30. A man expects to receive $12,000 in 10 years. How much is that money worth now, considering interest 
at 6% compounded semi-annually? 

We^are asked for the present value P which will amount to A - $12,000 in 10 years. Since there 
are k - 10 • 2 ^ 20 payment periods at interest rate r = .06/2 = .03 per period, the present value is 

P = $12,000(1 + .03)- 2" = $12,000(.55368) = $6644.16 
using the table on page 241. 

31. An investor has an annuity in which a payment of $500 is made at the end of each year. If interest 
is 4% compounded annually, what is the amount of the annuity after 20 years? 

Here r = .04, n = 20 and the amount is [see table on page 242], 



$500 



" (1 + .04)20 _ I 
.04 



$500(29.7781) = $14,889.05 



32. What is the present value of an annuity of $120 at the end of each 3 months for 12 years at 6% 
compounded quarterly? 

Here n = 4 • 12 = 48 payment periods, r = .06/4 =: .015 and the present value is 



$120 



.015 
using the table on page 243. 



L .0 



- (1.015) 



= $120(34.0426) = $4085.11 



TABLE 



1 



FOUR PLACE COMMON LOGARITHMS 

logioA^ or log A^ 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 




Proportion 


al Parts 


























I 2 


3 


4 


5 


6 


7 


8 9 


10 


0000 


0043 


0086 


0128 


0170 


0212 


0253 


0294 


0334 


0374 


4 8 


12 


17 


21 


25 


29 


33 37 


n 


0414 


0453 


0492 


0531 


0569 


0607 


0645 


0682 


0719 


0755 


4 8 


11 


15 


19 


23 


26 


30 34 


12 


0792 


0828 


0864 


0899 


0934 


0969 


1004 


1038 


1072 


1106 


3 7 


10 


14 


17 


21 


24 


28 31 


13 


1139 


1173 


1206 


1239 


1271 


1303 


1335 


1367 


1399 


1430 


3 6 


10 


13 


16 


19 


23 


26 29 


14 


1461 


1492 


1523 


1553 


1584 


1614 


1644 


1673 


1703 


1732 


3 6 


9 


12 


15 


18 


21 


24 27 


15 


1761 


1790 


1818 


1847 


1875 


1903 


1931 


1959 


1987 


2014 


3 6 


8 


n 


14 


17 


20 


22 25 


16 


2041 


2068 


2095 


2122 


2148 


2175 


2201 


2227 


2253 


2279 


3 5 


8 


11 


13 


16 


18 


21 24 


17 


2304 


2330 


2355 


2380 


2405 


2430 


2455 


2480 


2504 


2529 


2 5 


7 


10 


12 


15 


17 


20 22 


18 


2553 


2577 


2601 


2625 


2648 


2672 


2695 


2718 


2742 


2765 


2 5 


7 


9 


12 


14 


16 


19 21 


19 


2788 


2810 


2833 


2856 


2878 


2900 


2923 


2945 


2967 


2989 


2 4 


7 


9 


U 


13 


16 


18 20 


20 


3010 


3032 


3054 


3075 


3096 


3118 


3139 


3160 


3181 


3201 


2 4 


6 


8 


11 


13 


15 


17 19 


21 


3222 


3243 


3263 


3284 


3304 


3324 


3345 


3365 


3386 


3404 


2 4 


6 


8 


10 


12 


14 


16 18 


22 


3424 


3444 


3464 


3483 


3502 


3522 


3541 


3560 


3579 


3598 


2 4 


6 


8 


10 


12 


14 


15 17 


23 


3617 


3636 


3655 


3674 


3692 


3711 


3729 


3747 


3766 


3784 


2 4 


6 


7 


9 


11 


13 


15 17 


24 


3802 


3820 


3838 


3856 


3874 


3892 


3909 


3927 


3945 


3962 


2 4 


5 


7 


9 


11 


12 


14 16 


25 


3979 


3997 


4014 


4031 


4048 


4065 


4082 


4099 


4116 


4133 


2 3 


5 


7 


9 


10 


12 


14 15 


26 


4150 


4166 


4183 


4200 


4216 


4232 


4249 


4265 


4281 


4298 


2 3 


5 


7 


8 


10 


11 


13 15 


27 


4314 


4330 


4346 


4362 


4378 


4393 


4409 


4425 


4440 


4456 


2 3 


5 


6 


8 


9 


11 


13 14 


28 


4472 


4487 


4502 


4518 


4533 


4548 


4564 


4579 


4594 


4609 


2 3 


5 


6 


8 


9 


11 


12 14 


29 


4624 


4639 


4654 


4669 


4683 


4698 


4713 


4728 


4742 


4757 


1 3 


4 


6 


7 


9 


10 


12 13 


30 


4771 


4786 


4800 


4814 


4829 


4843 


4857 


4871 


4886 


4900 


1 3 


4 


6 


7 


9 


10 


11 13 


31 


4914 


4928 


4942 


4955 


4969 


4983 


4997 


5011 


5024 


5038 


1 3 


4 


6 


7 


8 


10 


11 12 


32 


5051 


5065 


5079 


5092 


5105 


6119 


5132 


5145 


5159 


5172 


1 3 


4 


5 


7 


8 


9 


11 12 


33 


5185 


5198 


6211 


5224 


5237 


5250 


5263 


5276 


5289 


5302 


1 3 


4 


6 


6 


8 


9 


10 12 


34 


5315 


5328 


5340 


5353 


5366 


5378 


5391 


5403 


5416 


5428 


1 3 


4 


5 


6 


8 


9 


10 11 


35 


6441 


5453 


5465 


5478 


5490 


5502 


5514 


5527 


5539 


5551 


1 2 


4 


5 


6 


7 


9 


10 U 


36 


5563 


5575 


5587 


5599 


5611 


5623 


5635 


5647 


6658 


5670 


1 2 


4 


5 


6 


7 


8 


10 11 


37 


5682 


5694 


5705 


5717 


5729 


5740 


5752 


5763 


6775 


5786 


1 2 


3 


5 


6 


7 


8 


9 10 


38 


5798 


5809 


5821 


5832 


5843 


5855 


5866 


5877 


5888 


5899 


1 2 


3 


5 


6 


7 


8 


9 10 


39 


5911 


5922 


5933 


5944 


5955 


5966 


5977 


5988 


5999 


6010 


1 2 


3 


4 


5 


7 


8 


9 10 


40 


6021 


6031 


6042 


6053 


6064 


6075 


6085 


6096 


6107 


6117 


1 2 


3 


4 


5 


6 


8 


9 10 


41 


6128 


6138 


6149 


6160 


6170 


6180 


6191 


6201 


6212 


6222 


1 2 


3 


4 


5 


6 


7 


8 9 


42 


6232 


6243 


6253 


6263 


6274 


6284 


6294 


6304 


6314 


6325 


1 2 


3 


4 


5 


6 


7 


8 9 


43 


6335 


6345 


6355 


6365 


6375 


6385 


6395 


6406 


6415 


6425 


1 2 


3 


4 


5 


6 


7 


8 9 


44 


6435 


6444 


6464 


6464 


6474 


6484 


6493 


6503 


6513 


6522 


1 2 


3 


4 


5 


6 


7 


8 9 


45 


6632 


6542 


6551 


6561 


6571 


6580 


6590 


6599 


6609 


6618 


1 2 


3 


4 


5 


6 


7 


8 9 


46 


6628 


6637 


6646 


6656 


6665 


6675 


6684 


6693 


6702 


6712 


1 2 


3 


4 


5 


6 


7 


7 8 


47 


6721 


6730 


6739 


6749 


6758 


6767 


6776 


6786 


6794 


6803 


1 2 


3 


4 


5 


5 


6 


7 8 


48 


6812 


6821 


6830 


6839 


6848 


6857 


6866 


6875 


6884 


6893 


1 2 


3 


4 


4 


5 


6 


7 8 


49 


6902 


6911 


6920 


6928 


6937 


6946 


6955 


6964 


6972 


6981 


1 2 


3 


4 


4 


5 


6 


7 8 


50 


6990 


6998 


7007 


7016 


7024 


7033 


7042 


7050 


7059 


7067 


1 2 


3 


3 


4 


5 


6 


7 8 


51 


7076 


7084 


7093 


7101 


7110 


7118 


7126 


7135 


7143 


7152 


1 2 


3 


3 


4 


5 


6 


7 8 


52 


7160 


7168 


7177 


7185 


7193 


7202 


7210 


7218 


7226 


7235 


1 2 


2 


3 


4 


5 


6 


7 7 


53 


7243 


7251 


7259 


7267 


7275 


7284 


7292 


7300 


7308 


7316 


1 2 


2 


3 


4 


6 


6 


6 7 


54 


7324 


7332 


7340 


7348 


7356 


7364 


7372 


7380 


7388 


7396 


1 2 


2 


3 


4 


5 


6 


6 7 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


1 2 


3 


4 


5 


6 


7 


8 9 



202 



Tabre 1 








FOUR PLACE COMMON LOGARITHMS 












(continued) 






logio A^ 


or I09 


A^ 
















N 





1 2 


3 


4 


5 6 


7 


8 


9 






Proportional Parts 
























1 


2 


3 


4 


5 


6 


7 


8 


9 


55 


7404 


7412 7419 


7427 


7435 


7443 7451 


7459 


7466 


7474 




2 


2 


3 


4 


5 


5 


6 


7 


56 


7482 


7490 7497 


7505 


7513 


7520 7528 


7536 


7643 


7551 




2 


2 


3 


4 


5 


5 


6 


7 


57 


7559 


7566 7574 


7582 


7589 


7597 7604 


7612 


7619 


7627 




2 


2 


3 


4 


5 


5 


6 


7 


58 


7634 


7642 7649 


7657 


7664 


7672 7679 


7686 


7694 


7701 






2 


3 


4 


4 


5 


6 


7 


59 


7709 


7716 7723 


7731 


7738 


7745 7752 


7760 


7767 


7774 






2 


3 


4 


4 


5 


6 


7 


60 


7782 


7789 7796 


7803 


7810 


7818 7825 


7832 


7839 


7846 






2 


3 


4 


4 


5 


6 


6 


61 


7853 


7860 7868 


7875 


7882 


7889 7896 


7903 


7910 


7917 






2 


3 


4 


4 


5 


6 


6 


62 


7924 


7931 7938 


7945 


7952 


7959 7966 


7973 


7980 


7987 






2 


3 


3 


4 


5 


6 


6 


63 


7993 


8000 8007 


8014 


8021 


8028 8035 


8041 


8048 


8055 






2 


3 


3 


4 


5 


5 


6 


64 


8062 


8069 8075 


8082 


8089 


8096 8102 


8109 


8116 


8122 






2 


3 


3 


4 


5 


5 


6 


65 


8129 


8136 8142 


8149 


8156 


8162 8169 


8176 


8182 


8189 






2 


3 


3 


4 


5 


5 


6 


66 


8195 


8202 8209 


8215 


8222 


8228 8235 


8241 


8248 


8254 






2 


3 


3 


4 


5 


5 


6 


67 


8261 


8267 8274 


8280 


8287 


8293 8299 


8306 


8312 


8319 






2 


3 


3 


4 


5 


5 


6 


68 


8325 


8331 8338 


8344 


8351 


8357 8363 


8370 


8376 


8382 






2 


3 


3 


4 


4 


5 


6 


69 


8388 


8395 8401 


8407 


8414 


8420 8426 


8432 


8439 


8445 






2 


2 


3 


4 


4 


5 


6 


70 


8451 


8457 8463 


8470 


8476 


8482 8488 


8494 


8500 


8506 






2 


2 


3 


4 


4 


5 


6 


71 


8513 


8519 8525 


8531 


8537 


8543 8549 


8555 


8561 


8567 






2 


2 


3 


4 


4 


5 


5 


72 


8573 


8579 8585 


8591 


8597 


8603 8609 


8615 


8621 


8627 






2 


2 


3 


4 


4 


5 


5 


73 


8633 


8639 8645 


8651 


8657 


8663 8669 


8675 


8681 


8686 






2 


2 


3 


4 


4 


5 


5 


74 


8692 


8698 8704 


8710 


8716 


8722 8727 


8733 


8739 


8745 






2 


2 


3 


4 


4 


5 


5 


75 


8751 


8756 8762 


8768 


8774 


8779 8785 


8791 


8797 


8802 






2 


2 


3 


3 


4 


5 


5 


76 


8808 


8814 8820 


8825 


8831 


8837 8842 


8848 


8854 


8859 






2 


2 


3 


3 


4 


5 


5 


77 


8865 


8871 8876 


8882 


8887 


8893 8899 


8904 


8910 


8915 






2 


2 


3 


3 


4 


4 


5 


78 


8921 


8927 8932 


8938 


8943 


8949 8954 


8960 


8965 


8971 






2 


2 


3 


3 


4 


4 


5 


79 


8976 


8982 8987 


8993 


8998 


9004 9009 


9015 


9020 


9025 






2 


2 


3 


3 


4 


4 


5 


80 


9031 


9036 9042 


9047 


9053 


9058 9063 


9069 


9074 


9079 






2 


2 


3 


3 


4 


4 


5 


81 


9085 


9090 9096 


9101 


9106 


9112 9117 


9122 


9128 


9133 






2 


2 


3 


3 


4 


4 


5 


82 


9138 


9143 9149 


9154 


9159 


9165 9170 


9175 


9180 


9186 






2 


2 


3 


3 


4 


4 


5 


83 


9191 


9196 9201 


9206 


9212 


9217 9222 


9227 


9232 


9238 






2 


2 


3 


3 


4 


4 


5 


84 


9243 


9248 9253 


9258 


9263 


9269 9274 


9279 


9284 


9289 






2 


2 


3 


3 


4 


4 


5 


85 


9294 


9299 9304 


9309 


9315 


9320 9325 


9330 


9335 


9340 






2 


2 


3 


3 


4 


4 


5 


86 


9345 


9350 9355 


9360 


9365 


9370 9375 


9380 


9385 


9390 






2 


2 


3 


3 


4 


4 


5 


87 


9395 


9400 9405 


9410 


9415 


9420 9425 


9430 


9435 


9440 









2 


2 


3 


3 


4 


4 


88 


9445 


9450 9455 


9460 


9465 


9469 9474 


9479 


9484 


9489 









2 


2 


3 


3 


4 


4 


89 


9494 


9499 9504 


9509 


9513 


9518 9523 


9528 


9533 


9538 









2 


2 


3 


3 


4 


4 


90 


9542 


9547 9552 


9557 


9562 


9566 9571 


9576 


9581 


9686 









2 


2 


3 


3 


4 


4 


91 


9590 


9595 9600 


9605 


9609 


9614 9619 


9624 


9628 


9633 









2 


2 


3 


3 


4 


4 


92 


9638 


9643 9647 


9652 


9657 


9661 9666 


9671 


9675 


9680 









2 


2 


3 


3 


4 


4 


93 


9685 


9689 9694 


9699 


9703 


9708 9713 


9717 


9722 


9727 









2 


2 


3 


3 


4 


4 


94 


9731 


9736 9741 


9745 


9750 


9754 9759 


9763 


9768 


9773 









2 


2 


3 


3 


4 


4 


95 


9777 


9782 9786 


9791 


9795 


9800 9805 


9809 


9814 


9818 









2 


2 


3 


3 


4 


4 


96 


9823 


9827 9832 


9836 


9841 


9845 9850 


9854 


9859 


9863 









2 


2 


3 


3 


4 


4 


97 


9868 


9872 9877 


9881 


9886 


9890 9894 


9899 


9903 


9908 









2 


2 


3 


3 


4 


4 


98 


9912 


9917 9921 


9926 


9930 


9934 9939 


9943 


9948 


9952 









2 


2 


3 


3 


4 


4 


99 


9956 


9961 9965 


9969 


9974 


9978 9983 


9987 


9991 


9996 









2 


2 


3 


3 


3 


4 


N 





1 2 


3 


4 


5 6 


7 


8 


9 


1 


2 


3 


4 


5 


6 


7 


8 


9 



203 



TABLE 



2 



FOUR PLACE COMMON ANTILOGARITHMS 



10^ or antilog p 



p 





1 


2 


3 


4 


5 


6 


7 


8 


9 






Proportional Parts 


























1 


2 


3 


4 


5 


6 


7 


8 9 


.00 


1000 


1002 


1005 


1007 


1009 


1012 


1014 


1016 


1019 


1021 














1 


2 


2 2 


.01 


1023 


1026 


1028 


1030 


1033 


1035 


1038 


1040 


1042 


1045 














1 


2 


2 2 


.02 


1047 


1050 


1052 


1054 


1057 


1059 


1062 


1064 


1067 


1069 














1 


2 


2 2 


.03 


1072 


1074 


1076 


1079 


1081 


1084 


1086 


1089 


1091 


1094 














1 


2 


2 2 


.04 


1096 


1099 


1102 


1104 


1107 


1109 


1112 


1114 


1117 


1119 













2 


2 


2 2 


.05 


1122 


1125 


1127 


1130 


1132 


1135 


1138 


1140 


1143 


1146 













2 


2 


2 2 


.06 


1148 


1151 


1153 


1156 


1159 


1161 


1164 


1X67 


1169 


1172 













2 


2 


2 2 


.07 


1175 


1178 


1180 


1183 


1186 


1189 


1191 


1194 


1197 


1199 













2 


2 


2 2 


.08 


1202 


1205 


1208 


1211 


1213 


1216 


1219 


1222 


1225 


1227 













2 


2 


2 3 


.09 


1230 


1233 


1236 


1239 


1242 


1245 


1247 


1250 


1253 


1256 













2 


2 


2 3 


.10 


1259 


1262 


1265 


1268 


1271 


1274 


1276 


1279 


1282 


1285 













2 


2 


2 3 


.11 


1288 


1291 


1294 


1297 


1300 


1303 


1306 


1309 


1312 


1315 











2 


2 


2 


2 3 


.12 


1318 


1321 


1324 


1327 


1330 


1334 


1337 


1340 


1343 


1346 











2 


2 


2 


2 3 


.13 


1349 


1352 


1355 


1358 


1361 


1365 


1368 


1371 


1374 


1377 











2 


2 


2 


3 3 


.14 


1380 


1384 


1387 


1390 


1393 


1396 


1400 


1403 


1406 


1409 











2 


2 


2 


3 3 


.15 


1413 


1416 


1419 


1422 


1426 


1429 


1432 


1435 


1439 


1442 











2 


2 


2 


3 3 


.16 


1445 


1449 


1452 


1455 


1459 


1462 


1466 


1469 


1472 


1476 











2 


2 


2 


3 3 


.17 


1479 


1483 


1486 


1489 


1493 


1496 


1500 


1503 


1507 


1510 











2 


2 


2 


3 3 


.18 


1514 


1517 


1521 


1524 


1528 


1531 


1535 


1538 


1542 


1545 











2 


2 


2 


3 3 


.19 


1549 


1552 


1556 


1560 


1563 


1567 


1570 


1574 


1578 


1581 











2 


2 


3 


3 3 


.20 


1585 


1589 


1592 


1596 


1600 


1603 


1607 


1611 


1614 


1618 











2 


2 


3 


3 3 


.21 


1622 


1626 


1629 


1633 


1637 


1641 


1644 


1648 


1652 


1656 









2 


2 


2 


3 


3 3 


.22 


1660 


1663 


1667 


1671 


1675 


1679 


1683 


1687 


1690 


1694 









2 


2 


2 


3 


3 3 


.23 


1698 


1702 


1706 


1710 


1714 


1718 


1722 


1726 


1730 


1734 









2 


2 


2 


3 


3 4 


.24 


1738 


1742 


1746 


1750 


1754 


1758 


1762 


1766 


1770 


1774 









2 


2 


2 


3 


3 4 


.25 


1778 


1782 


1786 


1791 


1795 


1799 


1803 


1807 


1811 


1816 









2 


2 


2 


3 


3 4 


.26 


1820 


1824 


1828 


1832 


1837 


1841 


1845 


1849 


1854 


1858 









2 


2 


3 


3 


3 4 


.27 


1862 


1866 


1871 


1875 


1879 


1884 


1888 


1892 


1897 


1901 









2 


2 


3 


3 


3 4 


.28 


1905 


1910 


1914 


1919 


1923 


1928 


1932 


1936 


1941 


1945 









2 


2 


3 


3 


4 4 


.29 


1950 


1954 


1959 


1963 


1968 


1972 


1977 


1982 


1986 


1991 









2 


2 


3 


3 


4 4 


.30 


1995 


2000 


2004 


2009 


2014 


2018 


2023 


2028 


2032 


2037 









2 


2 


3 


3 


4 4 


.31 


2042 


2046 


2051 


2056 


2061 


2065 


2070 


2075 


2080 


2084 









2 


2 


3 


3 


4 4 


.32 


2089 


2094 


2099 


2104 


2109 


2113 


2118 


2123 


2128 


2133 









2 


2 


3 


3 


4 4 


.33 


2138 


2143 


2148 


2153 


2158 


2163 


2168 


2173 


2178 


2183 









2 


2 


3 


3 


4 4 


.34 


2188 


2193 


2198 


2203 


2208 


2213 


2218 


2223 


2228 


2234 


1 




2 


2 


3 


3 


4 


4 5 


.35 


2239 


2244 


2249 


2254 


2259 


2265 


2270 


2275 


2280 


2286 


1 




2 


2 


3 


3 


4 


4 5 


.36 


2291 


2296 


2301 


2307 


2312 


2317 


2323 


2328 


2333 


2339 


1 




2 


2 


3 


3 


4 


4 5 


.37 


2344 


2350 


2355 


2360 


2366 


2371 


2377 


2382 


2388 


2393 


1 




2 


2 


3 


3 


4 


4 5 


.38 


2399 


2404 


2410 


2415 


2421 


2427 


2432 


2438 


2443 


2449 


1 




2 


2 


3 


3 


4 


4 5 


.39 


2455 


2460 


2466 


2472 


2477 


2483 


2489 


2495 


2500 


2506 


1 




2 


2 


3 


3 


4 


5 5 


.40 


2512 


2518 


2523 


2529 


2535 


2541 


2547 


2553 


2559 


2564 


1 




2 


2 


3 


4 


4 


5 5 


.41 


2570 


2576 


2582 


2588 


2594 


2600 


2606 


2612 


2618 


2624 


1 




2 


2 


3 


4 


4 


5 5 


.42 


2630 


2636 


2642 


2649 


2655 


2661 


2667 


2673 


2679 


2685 


1 




2 


2 


3 


4 


4 


5 6 


.43 


2692 


2698 


2704 


2710 


2716 


2723 


2729 


2735 


2742 


2748 


1 




2 


3 


3 


4 


4 


5 6 


.44 


2754 


2761 


2767 


2773 


2780 


2786 


2793 


2799 


2805 


2812 


1 




2 


3 


3 


4 


4 


5 6 


.45 


2818 


2825 


2831 


2838 


2844 


2851 


2858 


2864 


2871 


2877 


1 




2 


3 


3 


4 


5 


5 6 


.46 


2884 


2891 


2897 


2904 


2911 


2917 


2924 


2931 


2938 


2944 


1 




2 


3 


3 


4 


5 


5 6 


.47 


2951 


2958 


2965 


2972 


2979 


2985 


2992 


2999 


3006 


3013 


1 




2 


3 


3 


4 


5 


5 6 


.48 


3020 


3027 


3034 


3041 


3048 


3055 


3062 


3069 


3076 


3083 


1 




2 


3 


4 


4 


5 


6 6 


.49 


3090 


3097 


3105 


3112 


3119 


3126 


3133 


3141 


3148 


3155 


1 




2 


3 


4 


4 


5 


6 6 


P 





1 


2 


3 


4 


5 


6 


7 


8 


9 


1 


2 


3 


4 


5 


6 


7 


8 9 



204 



Table 2 

(continued) 



FOUR PLACE COMMON ANTILOG ARITHMS 

10^ or antilog p 



p 



.50 
.51 
.52 
.53 
.54 

.55 
.56 

.57 
.58 
.59 

.60 
.61 
.62 
.63 
.64 

.65 
.66 
.67 
.68 
.69 

.70 
.71 
.72 
.73 

.74 

.75 
.76 
.77 

.78 
.79 

.80 
.81 
.82 

.83 
.84 

.85 
.86 
.87 
.88 
.89 

.90 

.91 
.92 
.93 
.94 

.95 
.96 
.97 
.98 

.99 







3162 3170 3177 3184 3192 

3236 3243 3251 3258 3266 

3311 3319 3327 3334 3342 

3388 3396 3404 3412 3420 

3467 3475 3483 3491 3499 

3548 3556 3565 3573 3581 

3631 3639 3648 3656 3664 

3715 3724 3733 3741 3750 

3802 3811 3819 3828 3837 

8890 3899 3908 3917 3926 



3981 3990 3999 

4074 4083 4093 

4169 4178 4188 

4266 4276 4285 

4365 4375 4385 

4467 4477 4487 

4571 4581 4592 

4677 4688 4699 

4786 4797 4808 

4898 4909 4920 

5012 5023 5035 

5129 5140 5152 

5248 5260 5272 

5370 5383 5395 

5495 5508 5521 

5623 5636 5649 

5754 5768 5781 

5888 5902 5916 

6026 6039 6053 

6166 6180 6194 



6310 6324 6339 

6457 6471 6486 

6607 6622 6037 

6761 6776 6792 

6918 6934 6950 

7079 7096 7112 

7244 7261 7278 

7413 7430 7447 

7586 7603 7621 

7762 7780 7798 

7943 7962 7980 

8128 8147 8166 

8318 8337 8356 

8511 8531 8551 

8710 8730 8750 

8913 8933 8954 

9120 9141 9162 

9333 9354 9376 

9550 9572 9594 

9772 9795 9817 



4009 4018 

4102 4111 

4198 4207 

4295 4305 

4395 4406 

4498 4508 

4603 4613 

4710 4721 

4819 4831 

4932 4943 

5047 5058 

5164 5176 

5284 5297 

5408 5420 

5534 5546 

5662 5675 

5794 5808 

5929 5943 

6067 6081 

6209 6223 

6353 6368 

6501 6516 

6653 6668 

6808 6823 

6966 6982 

7129 7145 
7295 7311 
7464 7482 
7638 7656 
7816 7834 

7998 8017 

8185 8204 

8375 8395 

8570 8590 

8770 8790 

8974 8995 

9183 9204 

9397 9419 

9616 9638 

9840 9863 







3199 3206 3214 3221 3228 

3273 3281 3289 3296 3304 

3350 3357 3365 3373 3381 

3428 3436 3443 3451 3459 

3508 3516 3524 3532 3540 

3589 3597 3606 3614 3622 

3673 3681 3690 3698 3707 

3758 3767 3776 3784 3793 

3846 3855 3864 3873 3882 

3936 3945 3954 3963 3972 

4027 4036 4046 4055 4064 

4121 4130 4140 4150 4159 

4217 4227 4236 4246 4256 

4315 4325 4335 4345 4355 

4416 4426 4436 4446 4457 

4519 4529 4539 4550 4560 

4624 4634 4645 4656 4667 

4732 4742 4753 4764 4775 

4842 4853 4864 4875 4887 

4955 4966 4977 4989 5000 

5070 5082 

5188 5200 

5309 5321 

5433 5445 

5559 5572 



5689 
5821 
5957 
6095 
6237 

6383 
6531 
6683 
6839 
6998 

7161 
7328 
7499 
7674 
7852 

8035 
8222 
8414 
8610 
8810 

9016 
9226 
9441 
9661 
9886 



5702 
5834 
5970 
6109 
6252 

6397 
6546 
6699 
6855 
7015 

7178 
7345 
7516 
7691 

7870 

8054 
8241 
8433 
8630 
8831 

9036 
9247 
9462 
9683 
9908 



5093 5105 5117 

5212 5224 5236 

5333 5346 5358 

5458 5470 5483 

5585 5598 5610 

5715 5728 5741 

5848 5861 5875 

5984 5998 6012 

6124 6138 6152 

6266 6281 6295 



6412 
6561 
6714 
6871 
7031 

7194 
7362 
7534 
7709 
7889 

8072 
8260 
8453 
8650 
8851 

9057 
9268 
9484 
9705 
9931 



6427 

6577 
6730 

6887 
7047 

7211 

7379 
7551 

7727 
7907 

8091 
8279 
8472 
8670 

8872 

9078 
9290 
9506 

9727 
9954 



6442 
6592 
6745 
6902 
7063 

7228 
7396 
7568 
7745 
7925 

8110 
8299 
8492 
8690 
8892 

9099 
9311 
9528 
9750 
9977 

9 



Proportional Parts 
123456789 



2 3 4 4 5 6 



3 
3 
3 
3 
3 

3 
3 
3 
4 
4 

4 
4 
4 
4 
4 



2 4 

2 4 

2 4 

2 4 



12 4 
1 2 4 
1 2 4 
1 3 4 
1 3 4 

13 4 
1 3 4 
1 3 4 
1 3 4 
1 3 4 



4 
5 
5 
5 
5 

5 
5 
5 
5 
5 

6 
6 
6 
6 
6 

6 
6 
7 
7 



2 3 4 6 5 6 7 



4 5 6 6 

4 5 6 7 

4 5 6 7 

4 5 6 7 

4 5 6 7 



4 5 

4 5 

4 5 

4 6 

5 6 

5 6 

5 6 

5 6 

5 6 

5 6 

5 7 

5 7 

5 7 

6 7 
6 7 

6 7 9 10 12 13 

6 8 9 11 12 14 

6 8 9 11 12 14 

6 8 9 11 13 14 

6 8 10 11 13 15 

7 8 10 12 13 15 
7 8 10 12 13 15 
7 9 10 12 14 16 
7 9 11 12 14 16 
7 9 11 13 14 16 



6 


7 


8 


9 


6 


7 


9 


10 


7 


8 


9 


10 


7 


8 


9 


10 


7 


8 


9 


10 


7 


8 


9 


11 


7 


8 


10 


11 


7 


9 


10 


11 


8 


9 


10 


11 


8 


9 


10 


12 


8 


9 


10 


12 


8 


9 


11 


12 


8 


10 


11 


12 


8 


10 


11 


13 


9 


10 


11 


13 



2 5 7 



7 9 11 

8 9 11 
8 10 12 
8 10 12 
8 10 12 

8 10 12 

8 11 13 

9 11 13 
9 11 13 
9 11 14 



13 15 17 

13 15 17 

14 15 17 
14 16 18 

14 16 18 

15 17 19 
15 17 19 

15 17 20 

16 18 20 
16 18 20 



12 3 4 5 



205 



X 


0' 


10' 


20' 


30' 


40' 


50' 


0° 


.0000 


.0029 


.0058 


.0087 


.0116 


.0145 


1 


.0175 


.0204 


.0233 


.0262 


.0291 


.0320 


2 


.0349 


.0378 


.0407 


.0436 


.0465 


.0494 


3 


.0523 


.0552 


.0581 


.0610 


.0640 


.0669 


4 


.0698 


.0727 


.0756 


.0785 


.0814 


.0843 


5° 


.0872 


.0901 


.0929 


.0958 


.0987 


.1016 


6 


.1045 


.1074 


.1103 


.1132 


.1161 


.1190 


7 


.1219 


.1248 


.1276 


.1305 


.1334 


.1363 


8 


.1392 


.1421 


.1449 


.1478 


.1507 


.1536 


9 


.1564 


.1593 


.1622 


.1650 


.1679 


.1708 


10° 


.1736 


.1765 


.1794 


.1822 


.1851 


.1880 


11 


.1908 


.1937 


.1965 


.1994 


.2022 


.2051 


12 


.2079 


.2108 


.2136 


.2164 


.2193 


.2221 


13 


.2250 


.2278 


.2306 


.2334 


.2363 


.2391 


14 


.2419 


.2447 


.2476 


.2504 


.2532 


.2560 


15° 


.2588 


.2616 


.2644 


.2672 


.2700 


.2728 


16 


.2756 


.2784 


.2812 


.2840 


.2868 


.2896 


17 


.2924 


.2952 


.2979 


.3007 


.3035 


.3062 


18 


.3090 


.3118 


.3145 


.3173 


.3201 


.3228 


19 


.3256 


.3283 


.3311 


.3338 


.3365 


.3393 


20° 


.3420 


.3448 


.3475 


.3502 


.3529 


.3557 


21 


.3584 


.3611 


.3638 


.3665 


.3692 


.3719 


22 


.3746 


.3773 


.3800 


.3827 


.3854 


.3881 


23 


.3907 


.3934 


.3961 


.3987 


.4014 


.4041 


24 


.4067 


.4094 


.4120 


.4147 


.4173 


.4200 


25° 


.4226 


.4253 


.4279 


.4305 


.4331 


.4358 


26 


.4384 


.4410 


.4436 


.4462 


.4488 


.4514 


27 


.4540 


.4566 


.4592 


.4617 


.4643 


.4669 


28 


.4695 


.4720 


.4746 


.4772 


.4797 


.4823 


29 


.4848 


.4874 


.4899 


.4924 


.4950 


.4975 


30° 


.5000 


.5025 


.5050 


.5075 


.6100 


.5125 


31 


.5150 


.5175 


.5200 


.5225 


.5250 


.5275 


32 


.5299 


.5324 


.5348 


.5373 


.5398 


.5422 


33 


.5446 


.5471 


.5495 


.5519 


.5544 


.5568 


34 


.5592 


.5616 


.5640 


.5664 


.5688 


.5712 


35° 


.5736 


.5760 


.5783 


.5807 


.5831 


.5854 


36 


.5878 


.5901 


.5925 


.5948 


.5972 


.5995 


37 


.6018 


.6041 


.6065 


.6088 


.6111 


.6134 


38 


.6157 


.6180 


.6202 


.6225 


.6248 


.6271 


39 


.6293 


.6316 


.6338 


.6361 


.6383 


.6406 


40° 


.6428 


.6450 


.6472 


.6494 


.6517 


.6539 


41 


.6561 


.6583 


.6604 


.6626 


.6648 


.6670 


42 


.6691 


.6713 


.6734 


.6756 


.6777 


,6799 


43 


.6820 


.6841 


.6862 


.6884 


.6905 


.6926 


44 


.6947 


.6967 


.6988 


.7009 


.7030 


.7050 


45° 


.7071 


.7092 


.7112 


.7133 


.7153 


.7173 



X 


0' 


10' 


20' 


30' 


40' 


50' 


45° 


.7071 


.7092 


.7112 


.7133 


.7153 


.7173 


46 


.7193 


.7214 


.7234 


.7254 


.7274 


.7294 


47 


.7314 


.7333 


.7353 


.7373 


.7392 


.7412 


48 


.7431 


.7451 


.7470 


.7490 


.7509 


.7528 


49 


.7547 


.7566 


.7585 


.7604 


.7623 


.7642 


50" 


.7660 


.7679 


.7698 


.7716 


.7735 


.7753 


51 


.7771 


.7790 


.7808 


.7826 


.7844 


.7862 


52 


.7880 


.7898 


.7916 


.7934 


.7951 


.7969 


53 


.7986 


.8004 


.8021 


.8039 


.8056 


.8073 


54 


.8090 


.8107 


.8124 


.8141 


.8158 


.8175 


55° 


.8192 


.8208 


.8225 


.8241 


.8258 


.8274 


56 


.8290 


.8307 


.8323 


.8339 


.8355 


.8371 


57 


.8387 


.8403 


.8418 


.8434 


.8450 


.8465 


58 


.8480 


.8496 


.8511 


.8526 


.8542 


.8557 


59 


.8572 


.8587 


.8G01 


.8616 


.8631 


.8646 


60° 


.8660 


.8675 


.8689 


.8704 


.8718 


.8732 


61 


.8746 


.8760 


.8774 


.8788 


.8802 


.8816 


62 


.8829 


.8843 


.8857 


.8870 


.8884 


.8897 


63 


.8910 


.8923 


.8936 


.8949 


.8962 


.8975 


64 


.8988 


.9001 


.9013 


.9026 


.9038 


.9051 


65° 


.9063 


.9075 


.9088 


.9100 


.9112 


.9124 


66 


.9135 


.9147 


.9159 


.9171 


.9182 


.9194 


67 


.9205 


.9216 


.9228 


.9239 


.9250 


.9261 


68 


.9272 


.9283 


.9293 


.9304 


.9315 


.9325 


69 


.9336 


.9346 


.9356 


.9367 


.9377 


.9387 


70° 


.9397 


.9407 


.9417 


.9426 


.9436 


.9446 


71 


.9455 


.9465 


.9474 


.9483 


.9492 


.9502 


72 


.9511 


.9520 


.9528 


.9537 


.9546 


.9555 


73 


.9563 


.9572 


.9580 


.9588 


.9596 


.9605 


74 


.9613 


.9621 


.9628 


.9636 


.9644 


.9652 


75° 


.9659 


.9667 


.9674 


.9681 


.9689 


.9696 


76 


.9703 


.9710 


.9717 


.9724 


.9730 


.9737 


77 


.9744 


.9750 


.9757 


.9763 


.9769 


.9775 


78 


.9781 


.9787 


.9793 


.9799 


.9805 


.9811 


79 


.9816 


.9822 


.9827 


.9833 


.9838 


.9843 


80° 


.9848 


.9853 


.9858 


.9863 


.9868 


.9872 


81 


.9877 


.9881 


.9886 


.9890 


.9894 


.9899 


82 


.9903 


.9907 


.9911 


.9914 


.9918 


.9922 


83 


.9925 


.9929 


.9932 


.9936 


.9939 


.9942 


84 


.9945 


.9948 


.9951 


.9954 


.9957 


.9959 


85° 


.9962 


.9964 


.9967 


.9969 


.9971 


.9974 


86 


.9976 


.9978 


.9980 


.9981 


.9983 


.9985 


87 


.9986 


.9988 


.9989 


.9990 


.9992 


.9993 


88 


.9994 


.9995 


.9996 


.9997 


.9997 


.9998 


89 


.9998 


.9999 


.9999 


1.0000 


1.0000 


1.0000 


90° 


1.0000 



206 



0' 

1 

2 
3 

4 

6 
7 
8 
9 

10 = 

11 

12 

13 

14 

15 = 

16 

17 

18 

19 

20° 

21 

22 

23 

24 

25° 

26 

27 

28 

29 

30° 

31 

32 

33 

34 

35° 

36 

37 

38 

39 

40° 

41 

42 

43 

44 



0' 



10' 



20' 



30' 



1.0000 1.0000 1.0000 1.0000 

.9998 .9998 .9997 .9997 

.9994 .9993 .9992 .9990 

.9986 .9985 .9983 .9981 

.9976 .9974 .9971 .9969 



.9962 
.9945 
.9926 
.9903 
.9877 

.9848 
.9816 
.9781 
.9744 
.9703 

.9659 
.9613 
.9563 
.9611 
.9455 

.9397 
.9336 
.9272 
.9205 
.9135 

.9063 
.8988 
.8910 
.8829 
.8746 

.8660 
.8572 
.8480 
.8387 
.8290 

.8192 
.8090 
.7986 
.7880 
.7771 

.7660 
.7547 
.7431 
.7314 
.7193 



.9959 
.9942 
.9922 
.9899 
.9872 

.9843 
.9811 
.9775 
.9737 
.9696 

.9652 
.9605 
.9555 
.9502 
.9446 

.9387 
.9325 
.9261 
.9194 
.9124 

.9051 
.8975 
.8897 
.8816 
.8732 

.8646 
.8557 
.8465 
.8371 
.8274 

.8175 
.8073 
.7969 
.7862 
.7753 

.7642 
.7528 
.7412 
.7294 
.7173 



.9957 
.9939 
.9918 
.9894 
.9868 

.9838 
.9805 
.9769 
.9730 
.9689 

.9644 
.9596 
.9546 
.9492 
.9436 

.9377 
.9315 
.9250 
.9182 
.9112 

.9038 
.8962 
.8884 
.8802 
.8718 

.8631 
.8542 
.8450 
.8355 
.8258 

.8158 
.8056 
.7951 
.7844 
.7735 

.7623 
.7509 
.7392 
.7274 
.7153 



.9954 
.9936 
.9914 
.9890 
.9863 

.9833 
.9799 
.9763 

.9724 
.9681 

.9630 
.9588 
.9537 
.9483 
.9426 

.9367 
.9304 
.9239 
.9171 
.9100 

.9026 
.8949 
.8870 
.8788 
.8704 

.8616 
.8526 
.8434 
.8339 
.8241 

.8141 
.8039 
.7934 
.7826 
.7716 

.7604 
.7490 
.7373 
.7254 
.7133 



40' 

.9999 
.9996 
.9989 
.9980 
.9967 

.9951 
.9932 
.9911 
.9886 
.9858 

.9827 
.9793 
.9757 
.9717 
.9674 

.9628 
.9580 
.9528 
.9474 
.9417 

.9356 
.9293 
.9228 
.9159 
.9088 

.9013 
.8936 
.8857 

.8774 
.8689 

.8601 
.8511 
.8418 
.8323 
.8225 

.8124 
.8021 
.7916 
.7808 
.7698 

.7585 
.7470 
.7353 
.7234 
.7112 



50' 

.9999 
.9995 
.9988 
.9978 
.9964 

.9948 
.9929 
.9907 
.9881 
.9853 

.9822 
.9787 
.9750 
.9710 
.9667 

.9621 
.9572 
.9520 
.9465 
.9407 

.9346 
.9283 
.9216 
.9147 
.9075 

.9001 
.8923 
.8843 
.8760 
.8675 

.8587 
.8496 
.8403 
.8307 
.8208 

.8107 
.8004 
.7898 
.7790 
.7679 

.7566 
.7451 
.7333 
.7214 
.7092 



45^ 



.7071 .7050 .7030 .7009 .6988 .6967 



45" 

46 

47 

48 

49 

50° 
51 
52 
53 

54 

55° 

56 

57 

58 

59 

60° 

61 

62 

63 

64 



75' 

76 

77 

78 

79 

80-^ 

81 

82 

83 

84 

85° 

86 

87 

88 

89 



90' 



0' 

.7071 
.6947 
.6820 
.6691 
.6561 

.6428 
.6293 
.6157 
.6018 
.5878 

.5736 
.5592 
.5446 
.5299 
.5150 

.5000 
.4848 
.4695 
.4540 
.4384 



10' 

.7050 
.6926 
.6799 
.6670 
.6539 

.6406 
.6271 
.6134 
.5995 
.5854 

.5712 
.5568 
.5422 
.5275 
.5125 

.4975 
.4823 
.4669 
.4514 
.4358 



20' 

.7030 
.6905 
.6777 
.6648 
.6517 

.6383 
.6248 
.6111 
.5972 
.5831 

.5688 
.5544 
.5398 
.5250 
.5100 

.4950 
.4797 
.4643 
.4488 
.4331 



30' 

.7009 
.6884 
.6756 
.6626 
.6494 

.6361 
.6225 
.6088 
.5948 
.5807 

.5664 
.5519 
.5373 
.5225 
.5075 

.4924 
.4772 
.4617 
.4462 
.4305 



40' 

.6988 
.6862 
.6734 
.6604 
.6472 

.6338 
.6202 
.6065 
.5925 
.5783 

.5640 
.5495 
.5348 
.5200 
.5050 

.4899 
.4746 
.4592 
.4436 
.4279 



.2588 
.2419 
.2250 
.2079 
.1908 

.1736 
.1564 
.1392 
.1219 
.1045 

.0872 
.0698 
.0523 
.0349 
.0175 



.2560 
.2391 
.2221 
.2051 
.1880 

.1708 
.1536 
.1363 
.1190 
.1016 

.0843 
.0669 
.0494 
.0320 
.0145 



.2532 

.2363 
.2193 
.2022 
.1851 

.1679 
.1507 
.1334 
.1161 
.0987 

.0814 
.0640 
.0465 
.0291 
.0116 



.2504 

.2334 
.2164 
.1994 
.1822 

.1650 
.1478 
.1305 
.1132 
.0958 

.0785 
.0610 
.0436 
.0262 
.0087 



.2476 
.2306 
.2136 
.1965 
.1794 

.1622 
.1449 
.1276 
.1103 
.0929 

.0756 
.0581 
.0407 
.0233 
.0058 



50' 

.6967 
.6841 
.6713 
.6583 
.6450 

.6316 
.6180 
.6041 
.5901 
.5760 

.5616 
.5471 
.5324 

.5175 
.5025 

.4874 
.4720 
.4566 
.4410 
.4253 



65° 


.4226 


.4200 


.4173 


.4147 


.4120 


.4094 


66 


.4067 


.4041 


.4014 


.3987 


.3961 


.3934 


67 


.3907 


.3881 


.3854 


.3827 


.3800 


.3773 


68 


.3746 


.3719 


.3692 


.3665 


.3638 


.3611 


69 


.3584 


.3557 


.3529 


.3502 


.3475 


.3448 


70° 


.3420 


.3393 


.3365 


.3338 


.3311 


.3283 


71 


.3256 


.3228 


.3201 


.3173 


.3145 


.3118 


72 


.3090 


.3062 


.3035 


.3007 


.2979 


.2952 


73 


.2924 


.2896 


.2868 


.2840 


.2812 


.2784 


74 


.2756 


.2728 


.2700 


.2672 


.2644 


.2616 



.2447 
.2278 
.2108 
.1937 
.1765 

.1593 
.1421 
.1248 
.1074 
.0901 

.0727 
.0552 
.0378 
.0204 
.0029 



.0000 



207 



0° 

1 

2 
3 
4 

5° 

6 

7 

8 

9 

10° 

11 

12 

13 

14 

15° 

16 

17 

18 

19 

20° 
21 
22 
23 

24 

25° 

26 

27 

28 

29 

30° 

31 

32 

33 

34 

35° 

36 

37 

38 

39 

40° 
41 
42 
43 

44 



45' 



0' 

.0000 
.0175 
.0349 
.0524 
.0699 

.0875 
.1051 
.1228 
.1405 
.1584 

.1763 
.1944 
.2126 
.2309 
.2493 

.2679 
.2867 
.3057 
.3249 
.3443 

.3640 
.3839 
.4040 

.4245 
.4452 

.4663 

.4877 
.5095 
.5317 
.5543 

.5774 
.6009 
.6249 
.6494 
.6745 

.7002 
.7265 

.7536 
.7813 
.8098 

.8391 
.8693 
.9004 
.9325 
.9657 



10' 

.0029 
.0204 
.0378 
.0553 
.0729 

.0904 
.1080 
.1257 
.1435 

.1614 

.1793 
.1974 
.2156 
.2339 
.2524 

.2711 
.2899 
.3089 
.3281 

.3476 

.3673 
.3872 

.4074 
.4279 
.4487 

.4699 
.4913 
.5132 

.5354 
.5581 

.5812 
.6048 
.6289 
.6536 
.6787 

.7046 
.7310 
.7581 
.7860 

.8146 

.8441 

.8744 
.9057 
.9380 
.9713 



20' 

.0058 
.0233 
.0407 
.0582 
.0758 

.0934 
.1110 
.1287 
.1465 
.1644 

.1823 
.2004 
.2186 
.2370 
.2555 

.2742 
.2931 
.3121 
.3314 
.3508 

.3706 
.3906 
.4108 
.4314 
.4522 

.4734 
.4950 
.5169 
.5392 
.5619 

.5851 
.6088 
.6330 
.6577 
.6830 

.7089 
.7355 
.7627 
.7907 
.8195 

.8491 
.8796 
.9110 
.9435 
.9770 



30' 

.0087 
.0262 
.0437 
.0612 
.0787 

.0963 
.1139 
.1317 
.1495 
.1673 

.1853 
.2035 
.2217 

.2401 
.2586 

.2773 
.2962 
.3153 
.3346 
.3541 

.3739 
.3939 

.4142 
.4348 
.4557 

.4770 
.4986 
.5206 
.5430 
.5668 

.5890 

.6128 
.6371 
.6619 
.6873 

.7133 
.7400 
.7673 
.7954 
.8243 

.8541 
.8847 
.9163 
.9490 
.9827 



40' 

.0116 
.0291 
.0466 
.0641 
.0816 

.0992 
.1169 
.1346 
.1524 
.1703 

.1883 
.2065 
.2247 
.2432 
.2617 

.2805 
.2994 
.3185 
.3378 
.3574 

.3772 
.3973 
.4176 
.4383 
.4592 

.4806 
.5022 
.5243 

.5467 
.5696 

.5930 
.6168 
.6412 
.6661 
.6916 

.7177 
.7445 
.7720 
.8002 
.8292 

.8591 

.8899 
.9217 
.9545 
.9884 



50' 

.0145 
.0320 
.0495 
.0670 
.0846 

.1022 
.1198 
.1376 
.1554 
.1733 

.1914 
.2095 
.2278 
.2462 
.2648 

.2836 
.3026 
.3217 
.3411 
.3607 

.3805 
.4006 
.4210 
.4417 
.4628 

.4841 
.5059 
.5280 
.5505 
.5735 

.5969 

.6208 
.6453 
.6703 

.6959 

.7221 

.7490 
.7766 
.8050 
.8342 

.8642 
.8952 
.9271 
.9601 
.9942 



1.0000 1.0058 1.0117 1.0176 1.0235 1.0295 



45° 

46 

47 

48 

49 

50° 

51 

52 

53 

54 

55° 

56 

57 

58 

59 

60° 

61 

62 

63 

64 

65° 

66 

67 

68 

69 

70° 

71 

72 

73 

74 

75° 

76 

77 

78 

79 

80° 

81 

82 

83 

84 

85° 

86 

87 

88 

89 



90° 



0' 

l.OOOO 
1.0355 
1.0724 
1.1106 
1.1504 

1.1918 
1.2349 
1.2799 
1.3270 
1.3764 

1.4281 
1.4826 
1.5399 
1.6003 
1.6643 

1.7321 
1.8040 
1.8807 
1.9626 
2.0503 

2.1445 
2.2460 
2.3559 

2.4751 
2.6051 

2.7475 
2.9042 
3.0777 
3.2709 

3.4874 

3.7321 

4.0108 
4.3315 
4.7046 
5.1446 

5.6713 

6.3138 
7.1154 
8.1443 
9.5144 



10' 

1.0058 
1.0416 
1.0786 
1.1171 
1.1571 

1.1988 
1.2423 
1.2876 
1.3351 
1.3848 

1.4370 
1.4919 
1.5497 
1.6107 
1.6753 

1.7437 
1.8165 
1.8940 
1.9768 
2.0655 

2.1609 
2.2637 

2.3750 
2.4960 
2.6279 

2.7725 
2.9319 
3.1084 
3.3052 
3.5261 

3.7760 
4.0611 

4.3897 
4.7729 
5.2257 

5.7694 

6.4348 
7.2687 
8.3450 
9.7882 



11.430 11.826 

14.301 14.924 

19.081 20.206 

28.636 31.242 

57.290 68.750 



20' 

1.0117 
1.0477 
1.0850 
1.1237 
1.1640 

1.2059 
1.2497 
1.2954 
1.3432 

1.3934 

1.4460 
1.5013 
1.5597 
1.6212 
1.6864 

1.7556 
1.8291 

1.9074 
1.9912 
2.0809 

2.1775 

2.2817 
2.3945 
2.5172 
2.6511 

2.7980 
2.9600 
3.1397 
3.3402 
3.5656 

3.8208 

4.1126 
4.4494 
4.8430 
5.3093 

5.8708 
6.5606 
7.4287 
8.5555 
10.078 

12.251 
15.605 
21.470 
34.368 
85.940 



30' 

1.0176 
1.0538 
1.0913 
1.1303 
1.1708 

1.2131 

1.2572 
1.3032 
1.3514 
1.4019 

1.4550 
1.5108 
1.5697 
1.6319 
1.6977 

1.7675 
1.8418 
1.9210 
2.0057 
2.0965 

2.1943 
2.2998 
2.4142 
2.5386 
2.6746 

2.8239 
2.9887 
3.1716 
3.3759 
3.6059 

3.8667 
4.1653 
4.5107 
4.9152 
5.3955 

5.9758 
6.6912 
7.5958 
8.7769 
10.385 

12.706 
16.350 
22.904 
38.188 
114.59 



40' 

1.0235 
1.0599 
1.0977 
1.1369 
1.1778 

1.2203 
1.2647 
1.3111 
1.3597 
1.4106 

1.4641 
1.5204 
1.5798 
1.6426 
1.7090 

1.7796 
1.8546 
1.9347 
2.0204 
2.1123 

2.2113 
2.3183 
2.4342 
2.5605 
2.6985 

2.8502 
3.0178 
3.2041 

3.4124 
3.6470 

3.9136 
4.2193 

4.5736 
4.9894 
5.4845 

6.0844 
6.8269 
7.7704 
9.0098 
10.712 



50' 

1.0295 

1.0661 
1.1041 
1.1436 

1.1847 

1.2276 
1.2723 
1.3190 
1.3680 
1.4193 

1.4733 
1.5301 
1.5900 
1.6534 

1.7205 

1.7917 
1.8676 
1.9486 
2.0353 
2.1283 

2.2286 
2.3369 
2.4545 
2.5826 
2.7228 

2.8770 
3.0475 
3.2371 
3.4495 
3.6891 

3.9617 

4.2747 
4.6382 
5.0658 
5.5764 

6.1970 
6.9682 
7.9530 
9.2553 
11.059 



13.197 13.727 

17.169 18.075 

24.542 26.432 

42.964 49.104 

171.89 343.77 



208 



0^ 

1 

2 
3 
4 

5° 

6 

7 

8 

9 

10' 

11 

12 

13 

14 

15 = 

16 

17 

18 

19 

20° 

21 

22 

23 

24 

25" 

26 

27 

28 

29 

30 '^ 

31 

32 

33 

34 

35° 

36 

37 

38 

39 

40° 
41 
42 
43 

44 



0' 



57.290 
28.636 
19.081 
14.301 



10' 

343.77 
49.104 
26.432 
18.075 
13.727 



11.430 11.059 

9.5144 9.2553 

8.1443 7.9530 

7.1154 6.9682 

6.3138 6.1970 



20' 

171.89 
42.964 
24.542 
17.169 
13.197 

10.712 
9.0098 
7.7704 
6.8269 
6.0844 



30' 

114.59 
38.188 
22.904 
1G.350 
12.706 

10.385 
8.7769 
7.5958 
6.6912 
5.9758 



40' 

85.940 
34.368 
21.470 
15.605 
12.251 



50' 

68.750 
31.242 
20.206 
14.924 
11.826 



10.078 9.7882 

8.5555 8.3450 

7.4287 7.2687 

6.5606 6.4348 

5.8708 5.7694 



5.6713 5,5764 5.4845 5.3955 5.3093 5.2257 

5.1446 5.0658 4.9894 4.9152 4.8430 4.7729 

4.7046 4.6382 4.5736 4.5107 4.4494 4.3897 

4.3315 4.2747 4.2193 4.1653 4.1126 4.0611 

4.0108 3.9617 3:9136 3.8667 3.8208 3.7760 



45° 



3.7321 


3.6891 


3.4874 


3.4495 


3.2709 


3.2371 


3.0777 


3.0475 


2.9042 


2.8770 


2.7475 


2.7228 


2.6051 


2.5826 


2.4751 


2.4545 


2.3559 


2.3369 


2.2460 


2.2286 



2.1445 
2.0503 
1.9626 
1.8807 
1.8040 

1.7321 
1.6643 
1.6003 
1.5399 
1.4826 

1.4281 
1.3764 
1.3270 
1.2799 
1.2349 

1.1918 
1.1504 
1.1106 
1.0724 
1.0355 



2.1283 
2.0353 
1.9486 
1.8676 
1.7917 

1.7205 
1.6534 
1.5900 
1.5301 
1.4733 

1.4193 
1.3680 
1.3190 
1.2723 
1.2276 

1.1847 
1.1436 
1.1041 
1.0661 
1.0295 



3.6470 
3.4124 
3.2041 
3.0178 
2.8502 

2.6985 
2.5605 
2.4342 
3.3183 
2.2113 

2.1123 
2.0204 
1.9347 
1.8546 
1.7796 

1.7090 
1.6426 
1.5798 
1.5204 
1.4641 



3.6059 
3.3759 
3.171G 
2.9887 
2.8239 

2.6746 
2.5386 
2.4142 
2.2998 
2.1943 

2.0965 
2.0057 
1.9210 
1.8418 
1.7675 

1.6977 
1.6319 
1.5697 
1.5108 
1.4550 



3.5656 
3.3402 
3.1397 
2.9600 
2.7980 

2.6511 
2.5172 
2.3945 
2.2817 
2.1775 

2.0809 
1.9912 
1.9074 
1.8291 
1.7556 

1.6864 
1.6212 
1.5597 
1.5013 
1.4460 



3.5261 
3.3052 
3.1084 
2.9319 
2.7725 

2.6279 
2.4960 
2.3750 
2.2637 
2.1609 

2.0655 
1.9768 
1.8940 
1.8165 
1.7437 

1.6753 
1.6107 
1.5497 
1.4919 
1.4370 



1.4106 1.4019 1.3934 1.3848 

1.3597 1.3514 1.3432 1.3351 

1.3111 1.3032 1.2954 1.2876 

1.2647 1.2572 1.2497 1.2423 

1.2203 1.2131 1.2059 1.1988 



1.1778 
1.1369 
1.0977 
1.0599 
1.0235 



1.1708 
1.1303 
1.0913 
1.0538 
1.0176 



1.1640 
1.1237 
1.0850 
1.0477 

1.0117 



1.1571 
1.1171 
1.0786 
1.0416 
1.0058 



1.0000 .9942 .9884 .9827 .9770 .9713 



45' 

46 

47 

48 

49 

50= 

51 

52 

53 

54 

55 '^ 

56 

57 

58 

59 

60' 
61 
62 
63 

64 

65 = 

66 

67 

68 

69 

70° 

71 

72 

73 

74 

75^ 

76 

77 

78 

79 

80° 
81 
82 
83 

84 

85 = 

86 

87 

88 

89 



0' 



10' 



20' 



30' 



40' 



50' 



1.000 .9942 .9884 .9827 .9770 .9713 

.9657 .9601 .9545 .9490 .9435 .9380 

.9325 .9271 .9217 .9163 .9110 .9057 

.9004 .8952 .8899 .8847 .8796 .8744 

.8693 .8642 .8591 .8541 .8491 .8441 

.8391 .8342 .8292 .8243 .8195 .8146 

.8098 .8050 .8002 .7954 .7907 .7860 

.7813 .7766 .7720 .7673 .7627 .7581 

.7536 .7490 .7445 .7400 .7355 .7310 

.7265 .7221 .7177 .7133 .7089 .7046 



.7002 .6959 

.6745 .6703 

.6494 .6453 

.6249 .6208 

.6009 .5969 

.5774 .5735 

.5543 .5505 

.5317 .5280 

.5095 .5059 

.4877 .4841 



.4663 
.4452 
.4245 
.4040 
.3839 

.3640 
.3443 
.3249 
.3057 
.2867 

.2679 
.2493 
.2309 
.2126 
.1944 

.1763 
.1584 
.1405 
.1228 
.1051 

.0875 
.0699 
.0524 
.0349 
.0175 



.4628 
.4417 
.4210 
.4006 
.3805 

.3607 
.3411 
.3217 
.3026 
.2836 

.2648 
.2462 
.2278 
.2095 
.1914 

.1733 
.1554 
.1376 
.1198 
.1022 

.0846 
.0670 
.0495 
.0320 
.0145 



.6916 
.6661 
.6412 
.6168 
.5930 

.5696 
.5467 
.5243 
.5022 
.4806 

.4592 
.4383 
.4176 
.3973 
.3772 

.3574 
.3378 
.3185 
.2994 
.2805 

.2617 
.2432 
.2247 
.2065 
.1883 

.1703 
.1524 
.1346 
.1169 
.0992 

.0816 
.0641 
.0466 
.0291 
.0116 



.6873 
.6619 
.6371 
.6128 
.5890 

.5658 
.5430 
.5206 
.4986 

.4770 

.4557 
.4348 
.4142 
.3939 
.3739 

.3541 
.3346 
.3153 
.2962 
.2773 

.2586 
.2401 
.2217 
.2035 
.1853 

.1673 
.1495 
.1317 
.1139 
.0963 

.0787 
.0612 
.0437 
.0262 
.0087 



.6830 
.6577 
.6330 
.6088 
.5851 

.5619 
.5392 
.5169 
.4950 
.4734 

.4522 
.4314 
.4108 
.3906 
.3706 

.3508 
.3314 
.3121 
.2931 
.2742 

.2555 
.2370 
.2186 
.2004 
.1823 

.1644 
.1465 
.1287 
.1110 
.0934 

.0758 
.0582 
.0407 
.0233 
.0058 



.6787 
.6536 
.6289 
.6048 
.5812 

.5581 
.5354 
.5132 
.4913 
,4699 

.4487 
.4279 
.4074 
.3872 
.3673 

.3476 
.3281 
.3089 
.2899 
.2711 

.2524 
.2339 
.2156 
.1974 
.1793 

.1614 
.1435 
.1257 
.1080 
.0904 

.0729 
.0553 
.0378 
.0204 
.0029 



90° .0000 



209 




X 


0' 


10' 


20' 


30' 


40' 


50' 


0° 


1.000 


1.000 


1.000 


1.000 


1.000 


1.000 


1 


1.000 


1.000 


1.000 


1.000 


1.000 


1.001 


2 


1.001 


1.001 


1.001 


1.001 


1.001 


1.001 


3 


1.001 


1.002 


1.002 


1.002 


1.002 


1.002 


4 


1.002 


1.003 


1.003 


1.003 


1.003 


1.004 


5° 


1.004 


1.004 


1.004 


1.005 


1.005 


1.005 


6 


1.006 


1.006 


1.006 


1.006 


1.007 


1.007 


7 


1.008 


1.008 


1.008 


1.009 


1.009 


1.009 


8 


1.010 


1.010 


1.011 


1.011 


1.012 


1.012 


9 


1.012 


1.013 


1.013 


1.014 


1.014 


1.015 


10" 


1.015 


1.016 


1.016 


1.017 


1.018 


1.018 


11 


1.019 


1.019 


1.020 


1.020 


1.021 


1.022 


12 


1.022 


1.023 


1.024 


1.024 


1.025 


1.026 


13 


1.026 


1.027 


1.028 


1.028 


1.029 


1.030 


14 


1.031 


1.031 


1.032 


1.033 


1.034 


1.034 


15° 


1.035 


1.036 


1.037 


1.038 


1.039 


1.039 


16 


1.040 


1.041 


1.042 


1.043 


1.044 


1.045 


17 


1.046 


1.047 


1.048 


1.048 


1.049 


1.050 


18 


1.051 


1.052 


1.053 


1.054 


1.056 


1.057 


19 


1.058 


1.059 


1.060 


1.061 


1.062 


1.063 


20° 


1.064 


1.065 


1.066 


1.068 


1.069 


1.070 


21 


1.071 


1.072 


1.074 


1.075 


1.076 


1.077 


22 


1.079 


1.080 


1.081 


1.082 


1.084 


1.085 


23 


1.086 


1.088 


1.089 


1.090 


1.092 


1.093 


24 


1.095 


1.096 


1.097 


1.099 


1.100 


1.102 


25° 


1.103 


1.105 


1.106 


1.108 


1.109 


1.111 


26 


1.113 


1.114 


1.116 


1.117 


1.119 


1.121 


27 


1.122 


1.124 


1.126 


1.127 


1.129 


1.131 


28 


1.133 


1.134 


1.136 


1.138 


1.140 


1.142 


29 


1.143 


1.145 


1.147 


1.149 


1.151 


1.153 


30° 


1.155 


1.157 


1.159 


1.161 


1.163 


1.165 


31 


1.167 


1.169 


1.171 


1.173 


1.175 


1.177 


32 


1.179 


1.181 


1.184 


1.186 


1.188 


1.190 


33 


1.192 


1.195 


1.197 


1.199 


1.202 


1.204 


34 


1.206 


1.209 


1.211 


1.213 


1.216 


1.218 


35° 


1.221 


1.223 


1.226 


1.228 


1.231 


1.233 


36 


1.236 


1.239 


1,241 


1.244 


1.247 


1.249 


37 


1.252 


1.255 


1.258 


1.260 


1.263 


1.266 


38 


1.269 


1.272 


1.275 


1.278 


1.281 


1.284 


39 


1.287 


1.290 


1.293 


1.296 


1.299 


1.302 


40° 


1.305 


1.309 


1.312 


1.315 


1.318 


1.322 


41 


1.325 


1.328 


1.332 


1.335 


1.339 


1.342 


42 


1.346 


1.349 


1.353 


1.356 


1.360 


1.364 


43 


1.367 


1.371 


1.375 


1.379 


1.382 


1.386 


44 


1.390 


1.394 


1.398 


1.402 


1.406 


1.410 


45° 


1.414 


1.418 


1.423 


1.427 


1.431 


1.435 



X 


0' 


10' 


20' 


30' 


40' 


50' 


45° 


1.414 


1.418 


1.423 


1.427 


1.431 


1.435 


46 


1.440 


1.444 


1.448 


1.453 


1.457 


1.462 


47 


1.466 


1.471 


1.476 


1.480 


1.485 


1.490 


48 


1.494 


1.499 


1.504 


1.509 


1.514 


1.519 


49 


1.524 


1.529 


1.535 


1.510 


1.545 


1.550 


50° 


1.556 


1.561 


1.567 


1.572 


1.578 


1.583 


51 


1.589 


1.595 


1.601 


1.606 


1.612 


1.618 


52 


1.624 


1.630 


1.636 


1.643 


1.649 


1.655 


53 


1.662 


1.668 


1.675 


1.681 


1.688 


1.695 


54 


1.701 


1.708 


1.715 


1.722 


1.729 


1.736 


55° 


1.743 


1.751 


1.758 


1.766 


1.773 


1.781 


56 


1.788 


1.796 


1.804 


1.812 


1.820 


1.828 


57 


1.836 


1.844 


1.853 


1.861 


1.870 


1.878 


58 


1.887 


1.896 


1.905 


1.914 


1.923 


1.932 


59 


1.942 


1.951 


1.961 


1.970 


1.980 


1.990 


60° 


2.000 


2.010 


2.020 


2.031 


2.041 


2.052 


61 


2.063 


2.074 


2.085 


2.096 


2.107 


2.118 


62 


2.130 


2.142 


2.154 


2.166 


2.178 


2.190 


63 


2.203 


2.215 


2.228 


2.241 


2.254 


2.268 


64 


2.281 


2.295 


2.309 


2.323 


2.337 


2.352 


65° 


2.366 


2.381 


2.396 


2.411 


2.427 


2.443 


66 


2.459 


2.475 


2.491 


2.508 


2.525 


2.542 


67 


2.559 


2.577 


2.595 


2.613 


2.632 


2.650 


68 


2.669 


2.689 


2.709 


2.729 


2.749 


2.769 


69 


2.790 


2.812 


2.833 


2.855 


2.878 


2.901 


70° 


2.924 


2.947 


2.971 


2.996 


3.021 


3.046 


71 


3.072 


3.098 


3.124 


3.152 


3.179 


3.207 


72 


3.236 


3.265 


3.295 


3.326 


3.357 


3.388 


73 


3.420 


3.453 


3.487 


3.521 


3.556 


3.592 


74 


3.628 


3.665 


3.703 


3.742 


3.782 


3.822 


75° 


3.864 


3.906 


3.950 


3.994 


4.039 


4.086 


76 


4.134 


4.182 


4.232 


4.284 


4.336 


4.390 


77 


4.445 


4.502 


4.560 


4.620 


4.682 


4.745 


78 


4.810 


4.876 


4.945 


5.016 


5.089 


5.164 


79 


5.241 


5.320 


5.403 


5.487 


5.575 


5.665 


80° 


5.759 


5.855 


5.955 


6.059 


6.166 


6.277 


81 


6.392 


6.512 


6.636 


6.765 


6.900 


7.040 


82 


7.185 


7.337 


7.496 


7.661 


7.834 


8.016 


83 


8.206 


8.405 


8.614 


8.834 


9.065 


9.309 


84 


9.567 


9.839 


10.13 


10.43 


10.76 


11.10 


85° 


11.47 


11.87 


12.29 


12.75 


13.23 


13.76 


86 


14.34 


14.96 


15.64 


16.38 


17.20 


18.10 


87 


19.11 


20.23 


21.49 


22.93 


24.56 


26.45 


88 


28.65 


31.26 


34.38 


38.20 


42.98 


49.11 


89 


57.30 


68.76 


85.95 


114.6 


171.9 


343.8 


90° 


CO 



210 



0° 

1 

2 
3 
4 

5° 

6 
7 
8 
9 

10° 

11 

12 

13 

14 

16 
17 
18 
19 

20° 

21 

22 

23 

24 



0' 



57.30 
28.65 
19.11 
14.34 

11.47 
9.567 
8.206 
7.185 
6.392 

5.759 
5.241 
4.810 
4.445 
4.134 

3.864 
3.628 
3.420 
3.236 
3.072 

2.924 
2.790 
2.669 
2.559 
2.459 



10' 

343.8 
49.11 
26.45 
18.10 
13.76 

11.10 
9.309 
8.016 
7.040 
6.277 

5.665 
5.164 
4.745 
4.390 
4.086 

3.822 
3.592 
3.388 
3.207 
3.046 

2.901 
2.769 
2.650 
2.542 
2.443 



20' 

171.9 
42.98 
24.56 
17.20 
13.23 

10.76 
9.065 
7.834 
6.900 
6.166 

5.575 
5.089 
4.682 
4.336 
4.039 

3.782 
3.556 
3.357 
3.179 
3.021 

2.878 
2.749 
2.632 
2.525 
2.427 



30' 

114,6 
38.20 
22.93 
16.38 
12.75 

10.43 
8.834 
7.661 
6.765 
6.059 

5.487 
5.016 
4.620 
4.284 
3.994 

3.742 
3.521 
3.326 
3.152 
2.996 

2.855 

2.729 
2.613 
2.508 
2.411 



40' 

85.95 
34.38 
21.49 
15.64 
12.29 

10.13 

8.614 
7.496 
6.636 
5.955 

5.403 
4.945 
4.560 
4.232 
3.950 

3.703 
3.487 
3.295 
3.124 
2.971 

2.833 
2.709 
2.595 
2.491 
2.396 



50' 

68.76 
31.26 
20.23 
14.96 
11.87 

9.839 
8.405 
7.337 
6.512 
5.855 

5.320 
4.876 
4.502 
4.182 
3.906 

3.665 
3.453 
3.265 
3.098 
2.947 

2.812 
2.689 
2.577 
2.475 
2.381 



25° 


2.366 


2.352 


2.337 


2.323 


2.309 


2.295 


26 


2.281 


2.268 


2.254 


2.241 


2.228 


2.215 


27 


2.203 


2.190 


2.178 


2.166 


2.154 


2.142 


28 


2.130 


2.118 


2.107 


2.096 


2.085 


2.074 


29 


2.063 


2.052 


2.041 


2.031 


2.020 


2.010 


30° 


2.000 


1.990 


1.980 


1.970 


1.961 


1.951 


31 


1.942 


1.932 


1.923 


1.914 


1.905 


1.896 


32 


1.887 


1.878 


1.870 


1.861 


1.853 


1.844 


33 


1.836 


1.828 


1.820 


1.812 


1.804 


1.796 


34 


1.788 


1.781 


1.773 


1.766 


1.758 


1.751 


35° 


1.743 


1.736 


1.729 


1.722 


1.715 


1.708 


36 


1.701 


1.695 


1.688 


1.681 


1.675 


1.668 


37 


1.662 


1.655 


1.649 


1.643 


1.636 


1.630 


38 


1.624 


1.618 


1.612 


1.606 


1.601 


1.595 


39 


1.589 


1.583 


1.578 


1.572 


1.567 


1.561 


40° 


1.556 


1.550 


1.545 


1.540 


1.535 


1.529 


41 


1.524 


1.519 


1.514 


1.509 


1.504 


1.499 


42 


1.494 


1.490 


1.485 


1.480 


1.476 


1.471 


43 


1.466 


1.462 


1.457 


1.453 


1.448 


1.444 


44 


1.440 


1.435 


1.431 


1.427 


1.423 


1.418 


45° 


1.414 


1.410 


1.406 


1.402 


1.398 


1.394 



45° 

46 

47 

48 

49 

50° 

51 

52 

53 

54 

55° 

56 

57 

58 

59 

60° 

61 

62 

63 

64 

65° 

66 

67 

68 

69 

70° 

71 

72 

73 

74 

75° 

76 

77 

78 

79 

80° 

81 

82 

83 

84 

85° 

86 

87 

88 

89 

90° 



0' 

1.414 
1.390 
1.367 
1.346 
1.325 

1.305 
1.287 
1.269 
1.252 
1.236 

1.221 
1.206 
1.192 
1.179 
1.167 



1.103 
1.095 
1.086 
1.079 
1.071 

1.064 
1.058 
1.051 
1.046 
1.040 

1.035 
1.031 
1.026 
1.022 
1.019 

1.015 
1.012 
1.010 
1.008 
1.006 

1.004 
1.002 
1.001 
1.001 
1.000 

1.000 



10' 

1.410 
1.386 
1.364 
1.342 
1.322 

1.302 
1.284 
1.266 
1.249 
1.233 

1.218 
1.204 
1.190 
1.177 
1.165 



1.155 1.153 

1.143 1.142 

1.133 1.131 

1.122 1.121 

1.113 1.111 



1.102 
1.093 
1.085 
1.077 
1.070 

1.063 
1.057 
1.050 
1.045 
1.039 

1.034 
1.030 
1.026 
1.022 
1.018 

1.015 
1.012 
1.009 
1.007 
1.005 

1.004 
1.002 
1.001 
1.001 
1.000 



20' 

1.406 
1.382 
1.360 
1.339 
1.318 

1.299 
1.281 
1.263 
1.247 
1.231 

1.216 
1.202 
1.188 
1.175 
1.163 

1.151 
1.140 
1.129 
1.119 
1.109 

1.100 
1.092 
1.084 
1.076 
1.069 

1.062 
1.056 
1.049 
1.044 
1.039 

1.034 
1.029 
1.025 
1.021 
1.018 

1.014 
1.012 
1.009 
1.007 
1.005 

1.003 
1.002 
1.001 
1.000 
1.000 



30' 

1.402 
1.379 
1.356 
1.335 
1.315 

1.296 
1.278 
1.260 
1.244 
1.228 

1.213 
1.199 
1.186 
1.173 
1.161 

1.149 
1.138 
1.127 
1.117 
1.108 

1.099 
1.090 
1.082 
1.075 
1.068 

1.061 
1.054 
1.048 
1.043 
1.038 

1.033 
1.028 
1.024 
1.020 
1.017 

1.014 
1.011 
1.009 
1.006 
1.005 

1.003 
1.002 
1.001 
1.000 
1.000 



40' 

1.398 
1.375 
1.353 
1.332 
1.312 

1.293 
1.275 
1.258 
1.241 
1.226 

1.211 
1.197 
1.184 
1.171 
1.159 



1.097 
1.089 
1.081 
1.074 
1.066 

1.060 
1.053 
1.048 
1.042 
1.037 

1.032 
1.028 
1.024 
1.020 
1.016 

1.013 
1.011 
1.008 
1.006 
1.004 

1.003 
1.002 
1.001 
1.000 
1.000 



50' 

1.394 
1.371 
1.349 
1.328 
1.309 

1.290 
1.272 
1.255 
1.239 
1.223 

1.209 
1.195 
1.181 
1.169 
1.157 



1.147 1.145 

1.136 1.134 

1.126 1.124 

1.116 1.114 

1.106 1.105 



1.096 
1.088 
1.080 
1.072 
1.065 

1.059 
1.052 
1.047 
1.041 
1.036 

1.031 
1.027 
1.023 
1.019 
1.016 

1.013 
1.010 
1.008 
1.006 
1.004 

1.003 
1.002 
1.001 
1.000 
1.000 



211 



TABLE 



9 



NATURAL TRIGONOMETRIC FUNCTIONS (in radians) 



X 


Sin a; 


Cosx 


Tana: 


Cot a; 


Sec a; 


Cscx 


.00 


.00000 


1.00000 


.00000 


CO 


1.00000 


00 


.01 


.01000 


.99995 


.01000 


99.9967 


1.00005 


100.0017 


.02 


.02000 


.99980 


.02000 


49.9933 


1.00020 


50.0033 


.03 


.03000 


.99955 


.03001 


33.3233 


1.00045 


33.3383 


.04 


,03999 


.99920 


.04002 


24.9867 


1.00080 


25.0067 


.05 


.04998 


.99875 


.05004 


19.9833 


1.00125 


20.0083 


.06 


.05996 


.99820 


.06007 


16.6467 


1.00180 


16.6767 


.07 


.06994 


.99755 


.07011 


14.2624 


1.00246 


14.2974 


.08 


.07991 


.99680 


.08017 


12.4733 


1.00321 


12.5133 


.09 


.08988 


.99595 


.09024 


11.0811 


1.00406 


11.1261 


.10 


.09983 


.99500 


.10033 


9.9666 


1.00502 


10.0167 


.11 


.10978 


.99396 


.11045 


9.0542 


1.00608 


9.1093 


.12 


.11971 


.99281 


.12058 


8.2933 


1.00724 


8.3534 


.13 


.12963 


.99156 


.13074 


7.6489 


1.00851 


7.7140 


.14 


.13954 


.99022 


.14092 


7.0961 


1.00988 


7.1662 


.15 


.14944 


.98877 


.15114 


6.6166 


1.01136 


6.6917 


.16 


.16932 


.98723 


.16188 


6.1966 


1.01294 


6.2767 


.17 


.16918 


.98558 


.17166 


5.8256 


1.01463 


5.9108 


.18 


.17903 


.98384 


.18197 


6.4954 


1.01642 


5.5857 


.19 


.18886 


.98200 


.19232 


6.1997 


1.01833 


5.2950 


.20 


.19867 


.98007 


.20271 


4.9332 


1.02034 


5.0335 


.21 


.20846 


.97803 


.21314 


4.6917 


1.02246 


4.7971 


.22 


.21823 


.97590 


.22362 


4.4719 


1.02470 


4.5823 


.23 


.22798 


.97367 


.23414 


4.2709 


1.02705 


4.3864 


.24 


.23770 


.97134 


.24472 


4.0864 


1.02951 


4.2069 


.25 


.24740 


.96891 


.25534 


3.9163 


1.03209 


4.0420 


.26 


.25708 


.96639 


.26602 


3.7591 


1.03478 


3.8898 


.27 


.26673 


.96377 


.27676 


3.6133 


1.03759 


3.7491 


.28 


.27636 


.96106 


.28755 


3.4776 


1.04052 


3.6185 


.29 


.28595 


.95824 


.29841 


3.3511 


1.04358 


3.4971 


.30 


.29552 


.95534 


.30934 


3.2327 


1.04675 


3.3839 


.31 


.30506 


.95233 


.32033 


3.1218 


1.05005 


3.2781 


.32 


.31457 


.94924 


.33139 


3.0176 


1.05348 


3.1790 


.33 


.32404 


.94604 


.34252 


2.9195 


1.05704 


3.0860 


.34 


.33349 


.94275 


.35374 


2.8270 


1.06072 


2.9986 


.35 


.34290 


.93937 


.36503 


2.7395 


1.06454 


2.9163 


.36 


.35227 


.93590 


.37640 


2.6567 


1.06849 


2.8387 


.37 


.36162 


.93233 


.38786 


2.5782 


1.07258 


2.7654 


.38 


.37092 


.92866 


.39941 


2.5037 


1.07682 


2.6960 


.39 


.38019 


.92491 


.41105 


2.4328 


1.08119 


2.6303 


.40 


.38942 


.92106 


.42279 


2.3652 


1.08570 


2.5679 



212 



Toble 9 




(continued) 


NATURAL TRIGONOMETRIC FUNCTIONS (in radians) 




X 


Sin a; 


Cos a; 


Tan re 


Cot a; 


Sec a; 


Csca; 


- 




.40 


.38942 


.92106 


.42279 


2.3652 


1.0857 


2.5679 






.41 


.39861 


.91712 


.43463 


2.3008 


1.0904 


2.5087 






.42 


.40776 


.91309 


.44657 


2.2393 


1.0952 


2.4524 






.43 


.4X687 


.90897 


.45862 


2.1804 


1.1002 


2.3988 






.44 


.42594 


.90475 


.47078 


2.1241 


1.1053 


2.3478 






.45 


.43497 


.90045 


.48306 


2.0702 


1.1106 


2.2990 






.46 


.44395 


.89605 


.49545 


2.0184 


1.1160 


2.2525 






.47 


.45289 


.89157 


.50797 


1.9686 


1.1216 


2.2081 






.48 


.46178 


.88699 


.52061 


1.9208 


1.1274 


2.1655 






.49 


.47063 


.88233 


.53339 


1.8748 


1.1334 


2.1248 






.50 


.47943 


.87758 


.54630 


1.8305 


1.1395 


2.0858 






.51 


.48818 


.87274 


.55936 


1.7878 


1.1458 


2.0484 






.52 


.49688 


.86782 


.57256 


1.7465 


1.1523 


2.0126 






.53 


.50553 


.86281 


.58592 


1.7067 


1.1590 


1.9781 






.54 


.51414 


.85771 


.59943 


1.6683 


1.1659 


1.9450 






.55 


.52269 


.85252 


.61311 


1.6310 


1.1730 


1.9132 






.56 


.53119 


.84726 


.62695 


1.5950 


1.1803 


1.8826 






.57 


.53963 


.84190 


.64097 


1.5601 


1.1878 


1.8531 






.58 


.54802 


.83646 


.65517 


1.5263 


1.1955 


1.8247 






.59 


.55636 


.83094 


.66956 


1.4935 


1.2035 


1.7974 






.60 


.56464 


.82534 


.68414 


1.4617 


1.2116 


1.7710 






.61 


.67287 


.81965 


.69892 


1.4308 


1.2200 


1.7456 






.62 


.58104 


.81388 


.71391 


1.4007 


1.2287 


1.7211 






.63 


.58914 


.80803 


.72911 


1.3715 


1.2376 


1.6974 






.64 


.59720 


.80210 


.74454 


1.3431 


1.2467 


1.6745 






.65 


.60519 


.79608 


.76020 


1.3154 


1.2561 


1.6524 






.66 


.61312 


.78999 


.77610 


1.2885 


1.2658 


1.6310 






.67 


.62099 


.78382 


.79225 


1.2622 


1.2758 


1.6103 






.68 


.62879 


.77757 


.80866 


1.2366 


1.2861 


1.5903 






.69 


.63654 


.77125 


.82534 


1.2116 


1.2966 


1.5710 






.70 


.64422 


.76484 


.84229 


1.1872 


1.3075 


1.5523 






.71 


.65183 


.75836 


.85953 


1.1634 


1.3186 


1.5341 






.72 


.65938 


.75181 


.87707 


1.1402 


1.3301 


1.5166 






.73 


.66687 


.74517 


.89492 


1.1174 


1.3420 


1.4995 






.74 


.67429 


.73847 


.91309 


1.0952 


1.3542 


1.4830 






.75 


.68164 


.73169 


.93160 


1.0734 


1.3667 


1.4671 






.76 


.68892 


.72484 


.95045 


1.0521 


1.3796 


1.4515 






.77 


.69614 


.71791 


.96967 


1.0313 


1.3929 


1.4365 






.78 


.70328 


.71091 


.98926 


1.0109 


1.4066 


1.4219 






.79 


.71035 


.70385 


1.0092 


.99084 


1.4208 


1.4078 




- 


.80 


.71736 


.69671 


1.0296 


.97121 


1.4353 


1.3940 





213 



Table? 

(continued) 



NATURAL TRIGONOMETRIC FUNCTIONS (in radians) 



X 


Sinx 


Cos a; 


Tana: 


CotK 


Sec X 


Csc X 


.80 


.71736 


.69671 


1.0296 


.97121 


1.4353 


1.3940 


.81 


.72429 


.68950 


1.0505 


.95197 


1.4503 


1.3807 


.82 


.73115 


.68222 


1.0717 


.93309 


1.4658 


1.3677 


.83 


.73793 


.67488 


1.0934 


.91455 


1.4818 


1.3551 


.84 


.74464 


.66746 


1.1156 


.89635 


1.4982 


1.3429 


.85 


.75128 


.65998 


1.1383 


.87848 


1.5152 


1.3311 


.86 


.75784 


.65244 


1.1616 


.86091 


1.5327 


1.3195 


.87 


.76433 


.64483 


1.1853 


.84365 


1.5508 


1.3083 


.88 


.77074 


.63715 


1.2097 


.82668 


1.5695 


1.2975 


.89 


.77707 


.62941 


1.2346 


.80998 


1.5888 


1.2869 


.90 


.78333 


.62161 


1.2602 


.79355 


1.6087 


1.2766 


.91 


.78950 


.61375 


1.2864 


.77738 


1.6293 


1.2666 


.92 


.79560 


.60582 


1.3133 


.76146 


1.6507 


1.2569 


.93 


.80162 


.59783 


1.3409 


.74578 


1.6727 


1.2475 


.94 


.80756 


.58979 


1.3692 


.73034 


1.6955 


1.2383 


.95 


.81342 


.68168 


1.3984 


.71511 


1.7191 


1.2294 


.96 


.81919 


.57352 


1.4284 


.70010 


1.7436 


1.2207 


.97 


.82489 


.56530 


1.4592 


.68531 


1.7690 


1.2123 


.98 


.83050 


.55702 


1.4910 


.67071 


1.7953 


1.2041 


.99 


.83603 


.54869 


1.5237 


.65631 


1.8225 


1.1961 


1.00 


.84147 


.54030 


1.5574 


.64209 


1.8508 


1.1884 


1.01 


.84683 


.53186 


1.5922 


.62806 


1.8802 


1.1809 


1.02 


.85211 


.52337 


1.6281 


.61420 


1.9107 


1.1736 


1.03 


.85730 


.51482 


1.6652 


.60051 


1.9424 


1.1665 


1.04 


.86240 


.50622 


1.7036 


.58699 


1.9754 


1.1595 


1.05 


.86742 


.49757 


1.7433 


.57362 


2.0098 


1.1528 


1.06 


.87236 


.48887 


1.7844 


.56040 


2.0455 


1.1463 


1.07 


.87720 


.48012 


1.8270 


.54734 


2.0828 


1.1400 


1.08 


.88196 


.47133 


1.8712 


.53441 


2.1217 


1.1338 


1,09 


.88663 


.46249 


1.9171 


.52162 


2.1622 


1.1279 


1.10 


.89121 


.45360 


1.9648 


.50897 


2.2046 


1.1221 


1.11 


.89570 


.44466 


2.0143 


.49644 


2.2489 


1.1164 


1.12 


.90010 


.43568 


2.066O 


.48404 


2.2952 


1.1110 


1.13 


.90441 


.42666 


2.1198 


.47175 


2.3438 


1.1057 


1.14 


.90863 


.41759 


2.1759 


.45959 


2.3947 


1.1006 


1.15 


.91276 


.40849 


2.2346 


.44753 


2.4481 


1.0956 


1.16 


.91680 


.39934 


2.2958 


.43558 


2.5041 


1.0907 


1.17 


.92075 


.39015 


2.3600 


.42373 


2.5631 


1.0861 


1.18 


.92461 


.38092 


2.4273 


.41199 


2.6252 


1.0815 


1.19 


.92837 


.37166 


2.4979 


.40034 


2.6906 


1.0772 


1.20 


.93204 


.36236 


2.5722 


.38878 


2.7597 


1.0729 



214 



Table 9 

(continued) 



NATURAL TRIGONOMETRIC FUNCTIONS (in radians! 



X 


Sin X 


Cos a; 


Tana: 


Cot a; 


Sec a; 


Csca; 


1.20 


.93204 


.36236 


2.5722 


.38878 


2.7597 


1.07292 


1.21 


.93562 


.35302 


2.6503 


.37731 


2.8327 


1.06881 


1.22 


.93910 


.34365 


2.7328 


.36593 


2.9100 


1.06485 


1.23 


.94249 


.33424 


2.8198 


.35463 


2.9919 


1.06102 


1.24 


.94578 


.32480 


2.9119 


.34341 


3.0789 


1.05732 


1.25 


.94898 


.31532 


3.0096 


.33227 


3.1714 


1.05376 


1.26 


.95209 


.30582 


3.1133 


.32121 


3.2699 


1.05032 


1.27 


.95510 


.29628 


3.2236 


.31021 


3.3752 


1.04701 


1.28 


.95802 


.28672 


3.3414 


.29928 


3.4878 


1.04382 


1.29 


.96084 


.27712 


3.4672 


.28842 


3.6085 


1.04076 


1.30 


.96356 


.26750 


3.6021 


.27762 


3.7383 


1.03782 


1.31 


.96618 


.25785 


3.7471 


.26687 


3.8782 


1.03500 


1.32 


.96872 


.24818 


3.9033 


.25619 


4.0294 


1.03230 


1.33 


.97115 


.23848 


4.0723 


.24556 


4.1933 


1.02971 


1.34 


.97348 


.22875 


4.2556 


.23498 


4.3715 


1.02724 


1.35 


.97572 


.21901 


4.4552 


.22446 


4.5661 


1.02488 


1.36 


.97786 


.20924 


4.6734 


.21398 


4.7792 


1.02264 


1.37 


.97991 


.19945 


4,9131 


.20354 


5.0138 


1.02050 


1.38 


.98185 


.18964 


5.1774 


.19315 


5.2731 


1.01848 


1.39 


.98370 


.17981 


5.4707 


.18279 


5.5613 


1.01657 


1.40 


.98545 


.16997 


5.7979 


.17248 


5.8835 


1.01477 


1.41 


.98710 


.16010 


6.1654 


.16220 


6.2459 


1.01307 


1.42 


.98865 


.15023 


6.5811 


.15195 


6.6567 


1.01148 


1.43 


.99010 


.14033 


7.0555 


.14173 


7.1260 


1.00999 


1.44 


.99146 


.13042 


7.6018 


.13155 


7.6673 


1.00862 


1.45 


.99271 


.12050 


8.2381 


.12139 


8.2986 


1.00734 


1.46 


.99387 


.11057 


8.9886 


.11125 


9.0441 


1.00617 


1.47 


.99492 


.10063 


9.8874 


.10114 


9.9378 


1.00510 


1.48 


.99588 


.09067 


10.9834 


.09105 


11.0288 


1.00414 


1.49 


.99674 


.08071 


12.3499 


.08097 


12.3903 


1.00327 


1.50 


.99749 


.07074 


14.1014 


.07091 


14.1368 


1.00251 


1.51 


.99815 


.06076 


16.4281 


.06087 


16.4585 


1.00185 


1.52 


.99871 


.05077 


19.6695 


.05084 


19.6949 


1.00129 


1.53 


.99917 


.04079 


24.4984 


.04082 


24.5188 


1.00083 


1.54 


.99953 


.03079 


32.4611 


.03081 


32.4765 


1.00047 


1.55 


.99978 


.02079 


48.0785 


.02080 


48.0889 


1.00022 


1.56 


.99994 


.01080 


92.6205 


.01080 


92.6259 


1.00006 


1.57 


1.00000 


.00080 


1255.77 


.00080 


1255.77 


1.00000 


1.58 


.99996 


-.00920 


-108.649 


-.00920 


-108.654 


1.00004 


1.59 


.99982 


-.01920 


-52.0670 


-.01921 


-52.0766 


1.00018 


1.60 


.99957 


-.02920 


-34.2325 


-.02921 


-34.2471 


1.00043 



215 



TABLE 



10 



log sin X (a: in degrees and minutes) 

[subtract 10 from each entry] 



X 


0' 


10' 


20' 


30' 


40' 


50' 


0° 


_ 


7.4637 


7.7648 


7.9408 


8.0658 


8.1627 


1 


8.2419 


8.3088 


8.3668 


8.4179 


8.4637 


8.5050 


2 


8.5428 


8.5776 


8.6097 


8.6397 


8.6677 


8.6940 


3 


8.7188 


8.7423 


8.7645 


8.7857 


8.8059 


8.8251 


4 


8.8436 


8.8613 


8.8783 


8.8946 


8.9104 


8.925G 


5- 


8.9403 


8.9545 


8.9682 


8.9816 


8.9945 


9.0070 


6 


9.0192 


9.0311 


9.0426 


9.0539 


9.0648 


9.0755 


7 


9.0859 


9.0961 


9.1060 


9.1157 


9.1252 


9.1345 


8 


9.1436 


9.1525 


9.1612 


9.1697 


9.1781 


9.1863 


9 


9.1943 


9.2022 


9.2100 


9.2176 


9.2251 


9.2324 


10° 


9.2397 


9.2468 


9.2538 


9.2606 


9.2674 


9.2740 


11 


9.2806 


9.2870 


9.2934 


9.2997 


9.3058 


9.3119 


12 


9.3179 


9.3238 


9.3296 


9.3353 


9.3410 


9.3466 


13 


9.3521 


9.3575 


9.3629 


9.3682 


9.3734 


9.3786 


14 


9.3837 


9.3887 


9.3937 


9.3986 


9.4035 


9.4083 


16° 


9.4130 


9.4177 


9.4223 


9.4269 


9.4314 


9.4359 


16 


9.4403 


9.4447 


9.4491 


9.4533 


9.4576 


9.4618 


17 


9.4659 


9.4700 


9.4741 


9.4781 


9.4821 


9.4861 


18 


9.4900 


9.4939 


9.4977 


9.5015 


9.5052 


9.5090 


19 


9.5126 


9.5163 


9.5199 


9.5235 


9.5270 


9.5306 


20° 


9.5341 


9.5375 


9.5409 


9.5443 


9.5477 


9.5510 


21 


9.5543 


9.5576 


9.5609 


9.5641 


9.5673 


9.5704 


22 


9.5736 


9.5767 


9.5798 


9.5828 


9.5859 


9.5889 


23 


9.5919 


9.5948 


9.5978 


9.6007 


9.6036 


9.6065 


24 


9.6093 


9.6121 


9.6149 


9.6177 


9.6205 


9.6232 


25° 


9.6259 


9.6286 


9.6313 


9.6340 


9.6366 


9.6392 


26 


9.6418 


9.6444 


9.6470 


9.6495 


9.6521 


9.6546 


27 


9.6570 


9.6595 


9.6620 


9.6644 


9.6668 


9.6692 


28 


9.6716 


9.6740 


9.6763 


9.6787 


9.6810 


9.6833 


29 


9.6856 


9.6878 


9.6901 


9.6923 


9.6946 


9.6968 


30° 


9.6990 


9.7012 


9.7033 


9.7055 


9.7076 


9.7097 


31 


9.7118 


9.7139 


9.7160 


9.7181 


9.7201 


9.7222 


32 


9.7242 


9.7262 


9.7282 


9.7302 


9.7322 


9.7342 


33 


9.7361 


9.7380 


9.7400 


9.7419 


9.7438 


9.7457 


34 


9.7476 


9.7494 


9.7513 


9.7531 


9.7550 


9.7568 


35° 


9.7586 


9.7604 


9.7622 


9.7640 


9.7657 


9.7675 


36 


9.7692 


9.7710 


9.7727 


9.7744 


9.7761 


9.7778 


37 


9.7795 


9.7811 


9.7828 


9.7844 


9.7861 


9.7877 


38 


9.7893 


9.7910 


9.7926 


9.7941 


9.7957 


9.7973 


39 


9.7989 


9.8004 


9.8020 


9.8035 


9.8050 


9.8066 


40'' 


9.8081 


9.8096 


9.8111 


9.8125 


9.8140 


9.8155 


41 


9.8169 


9.8184 


9.8198 


9.8213 


9.8227 


9.8241 


42 


9.8255 


9.8269 


9.8283 


9.8297 


9.8311 


9.8324 


43 


9.8338 


9.8351 


9.8365 


9.8378 


9.8391 


9.8405 


44 


9.8418 


9.8431 


9.8444 


9.8457 


9.8469 


9.8482 


45'-' 


9.8495 


9.8507 


9.8520 


9.8532 


9.8545 


9.8557 



216 



Table 10 

(continued) 



lOQ sin X {x in degrees and minutes) 

[subtract 10 from each entry] 



X 


0' 


10' 


20' 


30' 


40' 


50' 


45° 


9.8495 


9.8507 


9.8520 


9.8532 


9.8545 


9.8557 


46 


9.8569 


9.8582 


9.8594 


9.8606 


9.8618 


9.8629 


47 


9.8641 


9.8653 


9.8665 


9.8676 


9.8688 


9.8699 


48 


9.8711 


9.8722 


9.8733 


9.8745 


9.8756 


9.8767 


49 


9.8778 


9.8789 


9.8800 


9.8810 


9.8821 


9.8832 


50° 


9.8843 


9.8853 


9.8864 


9.8874 


9.8884 


9.8895 


51 


9.8905 


9.8915 


9.8925 


9.8935 


9.8945 


3.8955 


52 


9.8965 


9.8975 


9.8985 


9.8995 


9.9004 


9.9014 


53 


9.9023 


9.9033 


9.9042 


9.9052 


9.9061 


9.9070 


54 


9.9080 


9.9089 


9.9098 


9.9107 


9.9116 


9.9125 


55° 


9.9134 


9.9142 


9.9151 


9.9160 


9.9169 


9.9177 


56 


9.9186 


9.9194 


9.9203 


9.9211 


9.9219 


9.9228 


57 


9.9236 


9.9244 


9.9252 


9.9260 


9.9268 


9.9276 


58 


9.9284 


9.9292 


9.9300 


9.9308 


9.9315 


9.9323 


59 


9.9331 


9.9338 


9.9346 


9.9353 


9.9361 


9.9368 


60° 


9.9375 


9.9383 


9.9390 


9.9397 


9.9404 


9.9411 


61 


9.9418 


9.9425 


9.9432 


9.9439 


9.9446 


9.9453 


62 


9.9459 


9.9466 


9.9473 


9.9479 


9.9486 


9.9492 


63 


9.9499 


9.9505 


9.9512 


9.9518 


9.9524 


9.9530 


64 


9.9537 


9.9543 


9.9549 


9.9555 


9.9561 


9.9567 


65° 


9.9573 


9.9579 


9.9584 


9.9590 


9.9596 


9.9602 


66 


9.9607 


9.9613 


9.9618 


9.9624 


9.9629 


9.9635 


67 


9.9640 


9.9646 


9.9651 


9.9656 


9.9661 


9.9667 


68 


9.9672 


9.9677 


9.9682 


9.9687 


9.9692 


9.9697 


69 


9.9702 


9.9706 


9.9711 


9.9716 


9.9721 


9.9725 


70° 


9.9730 


9.9734 


9.9739 


9.9743 


9.9748 


9.9752 


71 


9.9757 


9.9761 


9.9765 


9.9770 


9.9774 


9,9778 


72 


9.9782 


9.9786 


9.9790 


9.9794 


9.9798 


9.9802 


73 


9.9806 


9.9810 


9.9814 


9.9817 


9.9821 


9.9825 


74 


9.9828 


9.9832 


9.9836 


9.9839 


9.9843 


9.9846 


75° 


9.9849 


9.9853 


9.9856 


9.9859 


9.9863 


9.9866 


76 


9.9869 


9.9872 


9.9875 


9.9878 


9.9881 


9.9884 


77 


9.9887 


9.9890 


9.9893 


9.9896 


9.9899 


9.9901 


78 


9.9904 


9.9907 


9.9909 


9.9912 


9.9914 


9.9917 


79 


9.9919 


9.9922 


9.9924 


9.9927 


9.9929 


9.9931 


80° 


9.9934 


9.9936 


9.9938 


9.9940 


9.9942 


9.9944 


81 


9.9946 


9.9948 


9.9950 


9.9952 


9.9954 


9.9956 


82 


9.9958 


9.9959 


9.9961 


9.9963 


9.9964 


9.9966 


83 


9.9968 


9.9969 


9.9971 


9.9972 


9.9973 


9.9975 


84 


9.9976 


9.9977 


9.9979 


9.9980 


9.9981 


9.9982 


85° 


9.9983 


9.9985 


9.9986 


9.9987 


9.9988 


9.9989 


86 


9.9989 


9.9990 


9.9991 


9.9992 


9.9993 


9.9993 


87 


9.9994 


9.9995 


9.9995 


9.9996 


9.9996 


9.9997 


88 


9.9997 


9.9998 


9.9998 


9.9999 


9.9999 


9.9999 


89 


9.9999 


10.0000 


10.0000 


10.0000 


10.0000 


10.0000 


90° 


10.0000 













217 



TABLE 



n 



log COS X {x in degrees and minutes) 

[subtract 10 from each entry] 



X 


0' 


10' 


20' 


30' 


40' 


50' 


0^ 


10.0000 


10.0000 


10.0000 


10.0000 


10.0000 


10.0000 


1 


9.9999 


9.9999 


9.9999 


9.9999 


9.9998 


9.9998 


2 


9.9997 


9.9997 


9.9996 


9.9996 


9.9995 


9.9995 


3 


9.9994 


9.9993 


9.9993 


9.9992 


9.9991 


9.9990 


4 


9.9989 


9.9989 


9.9988 


9.9987 


9.9986 


9.9985 


5° 


9.9983 


9.9982 


9.9981 


9.9980 


9.9979 


9.9977 


6 


9.9976 


9.9975 


9.9973 


9.9972 


9.9971 


9.9969 


7 


9.9968 


9.9966 


9.9964 


9.9963 


9.9961 


9.9959 


8 


9.9958 


9.9956 


9.9954 


9.9952 


9.9950 


9.9948 


9 


9.9946 


9.9944 


9.9942 


9.9940 


9.9938 


9.9936 


10° 


9.9934 


9.9931 


9.9929 


9.9927 


9.9924 


9.9922 


11 


9.9919 


9.9917 


9.9914 


9.9912 


9.9909 


9.9907 


12 


9.9904 


9.9901 


9.9899 


9.9896 


9.9893 


9.9890 


13 


9.9887 


9.9884 


9.9881 


9.9878 


9.9875 


9.9872 


14 


9.9869 


9.9866 


9.9863 


9.9859 


9.9856 


9.9853 


15° 


9.9849 


9.9846 


9.9843 


9.9839 


9.9836 


9.9832 


16 


9.9828 


9.9825 


9.9821 


9.9817 


9.9814 


9.9810 


17 


9.9806 


9.9802 


9.9798 


9.9794 


9.9790 


9.9786 


18 


9.9782 


9.9778 


9.9774 


9.9770 


9.9765 


9.9761 


19 


9.9757 


9.9752 


9.9748 


9.9743 


9.9739 


9.9734 


20° 


9.9730 


9.9725 


9.9721 


9.9716 


9.9711 


9.9706 


21 


9.9702 


9.9697 


9.9692 


9.9687 


9.9682 


9.9677 


22 


9.9672 


9.9667 


9.9661 


9.9656 


9.9651 


9.9646 


23 


9.9640 


9.9635 


9.9629 


9.9624 


9.9618 


9.9613 


24 


9.9607 


9.9602 


9.9596 


9.9590 


9.9584 


9.9579 


25° 


9.9573 


9.9567 


9.9561 


9.9555 


9.9549 


9,9543 


26 


9.9537 


9.9530 


9.9524 


9.9518 


9.9512 


9.9505 


27 


9.9499 


9.9492 


9.9486 


9.9479 


9.9473 


9.9466 


28 


9.9459 


9.9453 


9.9446 


9.9439 


9.9432 


9.9425 


29 


9.9418 


9.9411 


9.9404 


9.9397 


9.9390 


9.9383 


30° 


9.9375 


9.9368 


9.9361 


9.9353 


9.9346 


9.9338 


31 


9.9331 


9.9323 


9.9315 


9.9308 


9.9300 


9.9292 


32 


9.9284 


9.9276 


9.9268 


9.9260 


9.9252 


9.9244 


33 


9.9236 


9.9228 


9.9219 


9.9211 


9.9203 


9.9194 


34 


9.9186 


9.9177 


9.9169 


9.9160 


9.9151 


9.9142 


35° 


9.9134 


9.9125 


9.9116 


9.9107 


9.9098 


9.9089 


36 


9.9080 


9.9070 


9.9061 


9.9052 


9.9042 


9.9033 


37 


9.9023 


9.9014 


9.9004 


9.8995 


9.8985 


9.8975 


38 


9.8965 


9.8955 


9.8945 


9.8935 


9.8925 


9.8915 


39 


9.8905 


9.8895 


9.8884 


9.8874 


9.8864 


9.8853 


40° 


9.8843 


9.8832 


9.8821 


9.8810 


9.8800 


9.8789 


41 


9.8778 


9.8767 


9.8756 


9.8745 


9.8733 


9.8722 


42 


9.8711 


9.8699 


9.8688 


9.8676 


9.8665 


9.8653 


43 


9.8641 


9.8629 


9.8618 


9.8606 


9.8594 


9.8582 


44 


9.8569 


9.8557 


9.8545 


9.8532 


9.8520 


9.8507 


45° 


9.8495 


9.8482 


9.8469 


9.8457 


9.8444 


9.8431 



218 



Table 1 1 

(continued) 



■Og cos X {x in degrees and minutes) 

[subtract 10 from each entryl 



X 


0' 


10' 


20' 


30' 


40' 


50' 


45° 


9.8495 


9.8482 


9.8469 


9,8457 


9.8444 


9.8431 


46 


9.8418 


9.8405 


9.8391 


9.8378 


9.8365 


9.8351 


47 


9.8338 


9.8324 


9.8311 


9.8297 


9.8283 


9.8269 


48 


9.8255 


9.8241 


9.8227 


9.8213 


9.8198 


9.8184 


49 


9.81G9 


9.8155 


9.8140 


9.8125 


9.8111 


9.8096 


50° 


9.8081 


9.8066 


9.8050 


9.8035 


9.8020 


9.8004 


51 


9.7989 


9.7973 


9.7957 


9.7941 


9.7926 


9.7910 


52 


9.7893 


9.7877 


9.7861 


9.7844 


9.7828 


9.7811 


53 


9.7795 


9.7778 


9.7761 


9.7744 


9.7727 


9.7710 


54 


9.7692 


9.7675 


9.7657 


9.7640 


9.7622 


9.7604 


55=^ 


9.7586 


9.7568 


9.7550 


9.7531 


9.7513 


9.7494 


56 


9.7476 


9.7457 


9.7438 


9.7419 


9.7400 


9.7380 


57 


9.7361 


9.7342 


9.7322 


9.7302 


9.7282 


9.7262 


58 


9.7242 


9.7222 


9.7201 


9.7181 


9.7160 


9.7139 


59 


9.7118 


9.7097 


9.7076 


9.7055 


9.7033 


9.7012 


60° 


9.6990 


9.6968 


9.6946 


9.6923 


9.6901 


9.6878 


61 


9.6856 


9.6833 


9.6810 


9.6787 


9.6763 


9.6740 


62 


9.6716 


9.6692 


9.6668 


9.6644 


9.6620 


9.6595 


63 


9.6570 


9.6546 


9.6521 


9.6495 


9.6470 


9.6444 


64 


9.6418 


9.6392 


9.6366 


9.6340 


9.6313 


9.6286 


65° 


9.6259 


9.6232 


9.6205 


9.6177 


9.6149 


9.6121 


66 


9.6093 


9.6065 


9.6036 


9.6007 


9.5978 


9.5948 


67 


9.5919 


9.5889 


9.5859 


9.5828 


9.5798 


9.5767 


68 


9.5736 


9.5704 


9.5673 


9.5641 


9.5609 


9.5576 


69 


9.5543 


9.5510 


9.5477 


9.5443 


9.5409 


9.5375 


70° 


9.5341 


9.5306 


9.5270 


9.5235 


9.5199 


9.5163 


71 


9.5126 


9.5090 


9.5052 


9.5015 


9.4977 


9.4939 


72 


9.4900 


9.4861 


9.4821 


9.4781 


9.4741 


9.4700 


73 


9.4659 


9.4618 


9.4576 


9.4533 


9.4491 


9.4447 


74 


9.4403 


9.4359 


9.4314 


9.4269 


9.4223 


9.4177 


75° 


9.4130 


9.4083 


9.4035 


9.3986 


9.3937 


9.3887 


76 


9.3837 


9.3786 


9.3734 


9.3682 


9.3629 


9.3575 


77 


9.3521 


9.3466 


9.3410 


9.3353 


9.3296 


9.3238 


78 


9.3179 


9.3119 


9.3058 


9.2997 


9.2934 


9.2870 


79 


9.2806 


9.2740 


9.2674 


9.2606 


9.2538 


9.2468 


80° 


9.2397 


9.2324 


9.2251 


9.2176 


9.2100 


9.2022 


81 


9.1943 


9.1863 


9.1781 


9.1697 


9.1612 


9.1525 


82 


9.1436 


9.1345 


9.1252 


9.1157 


9.1060 


9.0961 


83 


9.0859 


9.0756 


9.0648 


9.0539 


9.0426 


9.0311 


84 


9.0192 


9.0070 


8.9945 


8.9816 


8.9682 


8.9545 


85° 


8.9403 


8.9256 


8.9104 


8.8946 


8.8783 


8.8613 


86 


8.8436 


8.8251 


8.8059 


8.7857 


8.7645 


8.7423 


87 


8.7188 


8.6940 


8.6677 


8.6397 


8.6097 


8.5776 


88 


8.5428 


8.5050 


8.4637 


8.4179 


8.3668 


8.3088 


89 


8.2419 


8.1627 


8.0658 


7.9408 


7.7648 


7.4637 


90° 


- 













219 



TABLE 



12 



log ton X (a; in degrees and minutes) 

[subtract 10 from each entry] 



X 


0' 


10' 


20' 


30' 


40' 


50' 


0° 


— 


7.4637 


7.7648 


7.9409 


8.0658 


8.1627 


1 


8.2419 


8.3089 


8.3669 


8.4181 


8.4638 


8.5053 


2 


8.5431 


8.5779 


8.6101 


8.6401 


8.6682 


8.6945 


3 


8.7194 


8.7429 


8.7652 


8.7865 


8.8067 


8.8261 


4 


8.8446 


8.8624 


8.8795 


8.8960 


8.9118 


8.9272 


5° 


8.9420 


8.9563 


8.9701 


8.9836 


8.9966 


9.0093 


6 


9.0216 


9.0336 


9.0453 


9.0567 


9.0678 


9.0786 


7 


9.0891 


9.0995 


9.1096 


9.1194 


9.1291 


9.1385 


8 


9.1478 


9.1569 


9.1658 


9.1745 


9.1831 


9.1915 


9 


9.1997 


9.2078 


9.2158 


9.2236 


9.2313 


9.2389 


10° 


9.2463 


9.2536 


9.2609 


9.2680 


9.2750 


9.2819 


11 


9.2887 


9.2953 


9.3020 


9.3085 


9.3149 


9.3212 


12 


9.3275 


9.3336 


9.3397 


9.3458 


9.3517 


9.3576 


13 


9.3634 


9.3691 


9.3748 


9.3804 


9.3859 


9.3914 


14 


9.3968 


9.4021 


9.4074 


9.4127 


9.4178 


9.4230 


15° 


9.4281 


9.4331 


9.4381 


9.4430 


9.4479 


9.4527 


16 


9.4575 


9.4622 


9.4669 


9.4716 


9.4762 


9.4808 


17 


9.4853 


9.4898 


9.4943 


9.4987 


9.5031 


9.5075 


18 


9.5118 


9.5161 


9.5203 


9.5245 


9.52S7 


9.5329 


19 


9.5370 


9.5411 


9.5451 


9.5491 


9.5531 


9.5571 


20° 


9.5611 


9.5650 


9.5689 


9.5727 


9.5766 


9.5804 


21 


9.5842 


9.5879 


9.5917 


9.5954 


9.5991 


9.6028 


22 


9.6064 


9.6100 


9.6136 


9.6172 


9.6208 


9.6243 


23 


9.6279 


9.6314 


9.6348 


9.6383 


9.6417 


9.6452 


24 


9.6486 


9.6520 


9.6553 


9.6587 


9.6620 


9.6654 


25° 


9.6687 


9.6720 


9.6752 


9.6785 


9.6817 


9.6850 


26 


9.6882 


9.6914 


9.6946 


9.6977 


9.7009 


9.7040 


27 


9.7072 


9.7103 


9.7134 


9.7165 


9.7196 


9.7226 


28 


9.7257 


9.7287 


9.7317 


9.7348 


9.7378 


9.7408 


29 


9.7438 


9.7467 


9.7497 


9.7526 


9.7556 


9.7585 


30" 


9.7614 


9.7644 


9.7673 


9.7701 


9.7730 


9.7759 


31 


9.7788 


9.7816 


9.7845 


9.7873 


9.7902 


9.7930 


32 


9.7958 


9.7986 


9.8014 


9.8042 


9.8070 


9.8097 


33 


9.8125 


9.8153 


9.8180 


9.8208 


9.8235 


9.8263 


34 


9.8290 


9.8317 


9.8344 


9.8371 


9.8398 


9.8425 


35° 


9.8452 


9.8479 


9.8506 


9.8533 


9.8559 


9.8586 


36 


9.8613 


9.8639 


9.8666 


9.8692 


9.8718 


9.8745 


37 


9.8771 


9.8797 


9.8824 


9.8850 


9.8876 


9.8902 


38 


9.8928 


9.8954 


9.8980 


9.9006 


9.9032 


9.9058 


39 


9.9084 


9.9110 


9.9135 


9.9161 


9.9187 


9.9212 


40° 


9.9238 


9.9264 


9.9289 


9.9315 


9.9341 


9.9366 


41 


9.9392 


9.9417 


9.9443 


9.9468 


9.9494 


9.9619 


42 


9.9544 


9.9570 


9.9595 


9.9621 


9.9646 


9.9671 


43 


9.9697 


9.9722 


9.9747 


9.9772 


9.9798 


9.9823 


44 


9.9848 


9.9874 


9.9899 


9.9924 


9.9949 


9.9975 


45° 


10.0000 


10.0025 


10.0051 


10.0076 


10.0101 


10.0126 



220 



Table 12 

(continued) 



log ton X (a; in degrees and minutes) 

[subtract 10 from each entry] 



X 


0' 


10' 


20' 


30' 


40' 


50' 


45° 


10.0000 


10.0025 


10.0051 


10.0076 


10.0101 


10.0126 


46 


10.0152 


10.0177 


10.0202 


10.0228 


10.0253 


10.0278 


47 


10.0303 


10.0329 


10.0354 


10.0379 


10.0405 


10.0430 


48 


10.0456 


10.0481 


10.0506 


10.0532 


10.0557 


10.0583 


49 


10.0608 


10.0634 


10.0659 


10.0685 


10.0711 


10.0736 


50° 


10.0762 


10.0788 


10.0813 


10.0839 


10.0865 


10.0890 


51 


10.0916 


10.0942 


10.0968 


10.0994 


10.1020 


10.1046 


52 


10.1072 


10.1098 


10.1124 


10.1150 


10.1176 


10.1203 


53 


10.1229 


10.1255 


10.1282 


10.1308 


10.1334 


10.1361 


54 


10.1387 


10.1414 


10.1441 


10.1467 


10.1494 


10.1521 


55° 


10.1548 


10.1575 


10.1602 


10.1629 


10.1656 


10.1683 


56 


10.1710 


10.1737 


10.1765 


10.1792 


10.1820 


10.1847 


57 


10.1875 


10.1903 


10.1930 


10.1958 


10.1986 


10.2014 


58 


10.2042 


10.2070 


10.2098 


10.2127 


10.2155 


10.2184 


59 


10.2212 


10.2241 


10.2270 


10.2299 


10.2327 


10.2356 


60° 


10.2386 


10.2415 


10.2444 


10.2474 


10.2503 


10.2533 


61 


10.2562 


10.2592 


10.2622 


10.2652 


10.2683 


10.2713 


62 


10.2743 


10.2774 


10.2804 


10.2835 


10.2866 


10.2897 


63 


10.2928 


10.2960 


10.2991 


10.3023 


10.3054 


10.3086 


64 


10.3118 


10.3150 


10.3183 


10.3215 


10.3248 


10.3280 


65° 


10.3313 


10.3346 


10.3380 


10.3413 


10.3447 


10.3480 


66 


10.3514 


10.3548 


10.3583 


10.3617 


10.3652 


10.3686 


67 


10.3721 


10.3757 


10.3792 


10.3828 


10.3864 


10.3900 


68 


10.3936 


10.3972 


10.4009 


10.4046 


10.4083 


10.4121 


69 


10.4158 


10.4196 


10.4234 


10.4273 


10.4311 


10.4350 


70° 


10.4389 


10.4429 


10.4469 


10.4509 


10.4549 


10.4589 


71 


10.4630 


10.4671 


10.4713 


10.4755 


10.4797 


10.4839 


72 


10.4g82 


10.4925 


10.4969 


10.5013 


10.5057 


10.5102 


73 


10.5147 


10.5192 


10.5238 


10.5284 


10.5331 


10.5378 


74 


10.5425 


10.5473 


10.5521 


10.5570 


10.5619 


10.5669 


15° 


10.5719 


10.5770 


10.5822 


10.5873 


10.5926 


10.5979 


76 


10.6032 


10.6086 


10.6141 


10.6196 


10.6252 


10.6309 


77 


10.6366 


10.6424 


10.6483 


10.6542 


10.6603 


10.6664 


78 


10.6725 


10.6788 


10.6851 


10.6915 


10.6980 


10.7047 


79 


10.7113 


10.7181 


10.7250 


10.7320 


10.7391 


10.7464 


80° 


10.7537 


10.7611 


10.7687 


10.7764 


10.7842 


10.7922 


81 


10.8003 


10.8085 


10.8169 


10.8255 


10.8342 


10.8431 


82 


10.8522 


10.8615 


10.8709 


10.8806 


10.8904 


10.9005 


83 


10.9109 


10.9214 


10.9322 


10.9433 


10.9547 


10.9664 


84 


10.9784 


10.9907 


11.0034 


11.0164 


11.0299 


11.0437 


85° 


11.0580 


11.0728 


11.0882 


11.1040 


11.1205 


11.1376 


86 


11.1554 


11.1739 


11.1933 


11.2135 


11.2348 


11.2571 


87 


11.2806 


11.3055 


11.3318 


11.3599 


11.3899 


11.4221 


88 


11.4569 


11.4947 


11.5362 


11.5819 


11.6331 


11.6911 


89 


11.7581 


11.8373 


11.9342 


12.0591 


12.2352 


12.5363 


90° 















221 



TABLE 



13 



CONVERSrON OF RADIANS TO DEGREES, 
MINUTES AND SECONDS OR FRACTIONS OF DEGREES 



Radians 


Deg. 


Min. 


Sec. 


Fractions of 
Degrees 


1 


57° 


17' 


44.8" 


57.2958° 


2 


114° 


35' 


29.6" 


114.5916° 


3 


171° 


53' 


14.4" 


171.8873° 


4 


229° 


10' 


59.2" 


229.1831° 


5 


286° 


28' 


44.0" 


286.4789° 


6 


343° 


46' 


28.8" 


343.7747° 


7 
8 


401° 

458° 


4' 
21' 


13.6" 

58.4" 


401.0705° 

458.3662° 


9 


515° 


39' 


43.3" 


515.6620° 


10 


572° 


57' 


28.1" 


572.9578° 


.1 


5° 


43' 


46.5" 




.2 


11° 


27' 


33.0" 




.3 


17° 


11' 


19.4" 




.4 
.5 


22° 

28° 


65' 
38' 


5.9" 
52.4" 




.6 
.7 


34° 
40° 


22' 
6' 


38.9" 
25.4" 




.8 


45° 


50' 


11.8" 




.9 


51° 


33' 


58.3" 




.01 
.02 


0° 
1° 


34' 
8' 


22.6" 
45.3" 




.03 
.04 


1° 

2° 


43' 
17' 


7.9" 
30.6" 




.05 
.06 


2° 
3° 


51' 
26' 


53.2" 

15.9" 




.07 
.08 


4° 

4° 


0' 
35' 


38.5" 
1.2" 




.09 


5° 


9' 


23.8" 




.001 


0° 


3' 


26.3" 




.002 
.003 
.004 
.005 


0° 
0° 
0° 
0° 


6' 
10' 
13' 

17' 


52.5" 
18.8" 
45.1" 
11.3" 




.006 


0° 


20' 


37.6" 




.007 
.008 


0° 
0° 


24' 
27' 


3.9" 
30.1" 




.009 


0° 


30' 


56.4" 




.0001 


0° 


0' 


20.6" 




.0002 
.0003 


0° 
0° 


0' 
1' 


41.3" 

1.9" 




.0004 


0° 


1' 


22.5" 




.0005 


0° 


1' 


43.1" 




.0006 
.0007 


0° 
0° 


2' 
2' 


3.8" 
24.4" 




.0008 


0° 


2' 


45.0" 




.0009 


0° 


3' 


5.6" 





222 



TABLE 



14 



CONVERSION OF DEGREES, MINUTES 
AND SECONDS TO RADIANS 



Deg:rees 


Radians 


1^ 


.0174533 


2° 


.0349066 


3° 


.0523599 


4° 


.0698132 


5° 


.0872665 


00 


.1047198 


70 


.1221730 


8" 


.1396263 


9° 


.1570796 


10° 


.1745329 



Minutes 


Radians 


1' 


.00029089 


2' 


.00058178 


3' 


.00087266 


4' 


.00116355 


5' 


.00145444 


6' 


.00174533 


7' 


.00203622 


8' 


.00232711 


9' 


.00261800 


10' 


.00290888 



Seconds 


Radians 


X" 


.0000048481 


2" 


.0000096963 


3" 


.0000145444 


4" 


.0000193925 


5" 


.0000242407 


6" 


.0000290888 


7" 


.0000339370 


8" 


.0000387851 


9" 


.0000436332 


10" 


.0000484814 



223 



TABLE 



15 



NATURAL OR NAPIERIAN LOGARITHMS 

log^ X or In X 



X 







6 



1.0 
1.1 
1.2 
1.3 
1.4 

1.5 
1.6 
1.7 
1.8 
1.9 

2.0 
2.1 
2.2 
2.3 
2.4 

2.5 

2.6 
2.7 
2.8 
2.9 

3.0 
3.1 
3.2 
3.3 

3.4 

3.5 

3.6 
3.7 
3.8 
3.9 

4.0 
4.1 
4.2 
4.3 
4.4 

4.5 
4.6 
4.7 
4.8 
4.9 



.00000 
.09531 
.18232 
.26236 
.33647 

.40547 
.47000 
.53063 
.58779 
.64185 

.69315 
.74194 

.78846 
.83291 

.87547 

.91629 

.95551 

.99325 

1.02962 

1.06471 

1.09861 
1.13140 
1.16315 
1.19392 
1.22378 

1.25276 
1.28093 
1.30833 
1.33500 
1.36098 

1.38629 
1.41099 
1.43508 
1.45862 
1.48160 

1.50408 
1.52606 
1.54756 
1.56862 
1.58924 



.00995 
.10436 
.19062 
.27003 
.34359 

.41211 
.47623 
.53649 
.59333 
.64710 

.69813 
.74669 
.79299 
.83725 
.87963 

.92028 

.95935 

.99695 

1.03318 

1.06815 

1.10194 
1.13462 
1.16627 
1.19695 
1.22671 

1.25562 
1.28371 
1.31103 
1.33763 
1.36354 

1.38879 
1.41342 
1.43746 
1.46094 
1.48387 

1.50630 
1.52823 
1.54969 
1.57070 
1.59127 



.01980 
.11333 
.19885 
.27763 
.35066 

.41871 
.48243 
.54232 
.59884 
.65233 

.70310 
.75142 
.79751 
.84157 
.88377 

.92426 

.96317 

1.00063 

1.03674 

1.07158 

1.10526 
1.13783 
1.16938 
1.19996 
1.22964 

1.25846 
1.28647 
1.31372 
1.34025 
1.36609 

1.39128 
1.41585 
1.43984 
1.46326 
1.48614 

1.50851 
1.53039 
1.55181 

1.57277 
1.59331 



.02956 
.12222 
.20701 
.28518 
.35767 

.42527 
.48858 
.54812 
.60432 
.65752 

.70804 
.75612 
.80200 
.84587 
.88789 

.92822 

.96698 

1.00430 

1.04028 

1.07500 

1.10856 
1.14103 
1.17248 
1.20297 
1.23256 

1.26130 
1.28923 
1.31641 
1.34286 
1.36864 

1.39377 
1.41828 
1.44220 
1.46557 
1.48840 

1.51072 
1.53256 
1.55393 
1.57485 
1.59534 



.03922 
.13103 
.21511 
.29267 
.36464 

.43178 
.49470 
.55389 
.60977 
.66269 

.71295 
.76081 
.80648 
.85015 
.89200 

.93216 

.97078 

1.00796 

1.04380 

1.07841 

1.11186 
1.14422 
1.17557 
1.20597 
1.23547 

1.26413 
1.29198 
1.31909 
1.34547 
1.37118 

1.39624 
1.42070 
1.44456 
1.46787 
1.49065 

1.51293 
1.53471 
1.55604 
1.57691 
1.59737 



.04879 
.13976 
.22314 
.30010 
.37156 

.43825 
.50078 
.55962 
.61519 
.66783 

.71784 
.76547 
.81093 
.85442 
.89609 

.93609 

.97456 

1.01160 

1.04732 

1.08181 

1.11514 
1.14740 
1.17865 
1.20896 
1.23837 

1.26695 
1.29473 
1.32176 
1.34807 
1.37372 

1.39872 
1.42311 
1.44692 
1.47018 
1.49290 

1.51513 

1.53687 
1.55814 
1.57898 
1.59939 



.05827 
.14842 
.23111 
.30748 
.37844 

.44469 
.50682 
.56531 
.62058 

.67294 

.72271 
.77011 
.81536 
.85866 
.90016 

.94001 

.97833 

1.01523 

1.05082 

1.08519 

1.11841 
1.15057 
1.18173 
1.21194 
1.24127 

1.26976 
1.29746 
1.32442 
1.35067 
1.37624 

1.40118 
1.42552 
1.44927 
1.47247 
1.49515 

1.51732 
1.53902 
1.56025 
1.58104 
1.60141 



.06766 
.15700 
.23902 
.31481 
.38526 

.45108 
.51282 
.57098 

.62594 
.67803 

.72755 

.77473 
.81978 
.86289 
.90422 

.94391 

.98208 

1.01885 

1.05431 

1.08856 

1.12168 
1.15373 
1.18479 
1.21491 
1.24415 

1.27257 
1.30019 
1.32708 
1.35325 
1.37877 

1.40364 
1.42792 
1.45161 
1.47476 
1.49739 

1.51951 
1.54116 
1.56235 
1.58309 
1.60342 



.07696 
.16551 
.24686 
.32208 
.39204 

.45742 
.51879 
.57661 
.63127 
.68310 

.73237 
.77932 
.82418 
.86710 
.90826 

.94779 

.98582 

1.02245 

1.05779 

1.09192 

1.12493 
1.15688 
1.18784 
1.21788 
1.24703 

1.27536 
1.30291 
1.32972 
1.35584 
1.38128 

1.40610 
1.43031 
1.45395 
1.47705 
1.49962 

1.52170 
1.54330 
1.56444 
1.58515 
1.60543 



.08618 
.17395 
.25464 
.32930 
.39878 

.46373 
.52473 
.58222 
.63658 

.68813 

.73716 
.78390 
.82855 
.87129 
.91228 

.95166 

.98954 

1.02604 

1.06126 

1.09527 

1.12817 
1.16002 
1.19089 
1.22083 
1.24990 

1.27815 
1.30563 
1.33237 
1.35841 
1.38379 

1.40854 
1.43270 
1.45629 
1.47933 
1.50185 

1.52388 
1.54543 
1.56653 
1.58719 
1.60744 



In 10 = 2.30259 

2 In 10 = 4.60517 

3 In 10 = 6.90776 



4 In 10 = 


9.21034 


7 In 10 ^ 


^ 16.11810 


5 In 10 ^ 


11.51293 


8 In 10 ^ 


^ 18.42068 


6 In 10 = 


13.81551 


9 In 10 = 


= 20.72327 



224 



Table 15 

(continued) 



NATURAL OR NAPIERIAN LOGARITHMS 

log, X or In X 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


5.0 


1.60944 


1.61144 


1.61343 


1.61542 


1.61741 


1.61939 


1.62137 


1.62334 


1.62531 


1.62728 


5.1 


1.62924 


1.63120 


1.63315 


1.63511 


1.63705 


1.63900 


1.64094 


1.64287 


1.64481 


1.64673 


5.2 


1.64866 


1,65058 


1.65250 


1.65441 


1.65632 


1.65823 


1.66013 


1.66203 


1.66393 


1.66582 


6.3 


1.66771 


1.66959 


1.67147 


1.67335 


1.67523 


1.67710 


1.67896 


1.68083 


1.68269 


1.68455 


5.4 


1.68640 


1.68825 


1.69010 


1.69194 


1.69378 


1.69562 


1.69745 


1.69928 


1.70111 


1.70293 


5.6 


1.70475 


1.70656 


1.70838 


1.71019 


1.71199 


1.71380 


1.71560 


1.71740 


1.71919 


1.72098 


6.6 


1.72277 


1.72455 


1.72633 


1.72811 


1.72988 


1.73166 


1.73342 


1.73519 


1.73695 


1.73871 


6.7 


1.74047 


1.74222 


1.74397 


1.74572 


1.74746 


1.74920 


1.75094 


1.75267 


1.75440 


1.75613 


5.8 


1.75786 


1.75958 


1.76130 


1.76302 


1.76473 


1.76644 


1.76815 


1.76986 


1.77156 


1.77326 


5.9 


1.77495 


1.77665 


1.77834 


1.78002 


1.78171 


1.78339 


1.78507 


1.78675 


1.78842 


1.79009 


6.0 


1.79176 


1.79342 


1.79509 


1.79675 


1.79840 


1.80006 


1.80171 


1.80336 


1.80500 


1.80665 


6.1 


1.80829 


1.80993 


1.81156 


1.81319 


1.81482 


1.81645 


1.81808 


1.81970 


1.82132 


1.82294 


6.2 


1.82455 


1.82616 


1.82777 


1.82938 


1.83098 


1.83258 


1.8.3418 


1.83578 


1.83737 


1.83896 


6.3 


1.84055 


1.84214 


1.84372 


1.84530 


1.84688 


1.84845 


1.85003 


1.85160 


1.85317 


1.85473 


6.4 


1.85630 


1.85786 


1.85942 


1.86097 


1.86253 


1.86408 


1.86563 


1.86718 


1.86872 


1.87026 


6.5 


1.87180 


1.87334 


1.87487 


1.87641 


1.87794 


1.87947 


1.88099 


1.88251 


1.88403 


1.88555 


6.6 


1.88707 


1.88858 


1.89010 


1.89160 


1.89311 


1.89462 


1.89612 


1.89762 


1.89912 


1.90061 


6.7 


1.90211 


1.90360 


1.90509 


1.90658 


1.90806 


1.90954 


1.91102 


1.91250 


1.91398 


1.91545 


6.8 


1.91692 


1.91839 


1.91986 


1.92132 


1.92279 


1.92425 


1.92571 


1.92716 


1.92862 


1.93007 


6.9 


1.93152 


1.93297 


1.93442 


1.93586 


1.93730 


1.93874 


1.94018 


1.94162 


1.94306 


1.94448 


7.0 


1.94591 


1.94734 


1.94876 


1.95019 


1.95161 


1.95303 


1.95445 


1.95586 


1.95727 


1.95869 


7.1 


1.96009 


1.96150 


1.96291 


1.96431 


1.96571 


1.96711 


1.96851 


1.96991 


1.97130 


1.97269 


7.2 


1.97408 


1.97547 


1.97685 


1.97824 


1.97962 


1.98100 


1.98238 


1.98376 


1.98513 


1.98650 


7.3 


1.98787 


1.98924 


1.99061 


1.99198 


1.99334 


1.99470 


1.99606 


1.99742 


1.99877 


2.00013 


7.4 


2.00148 


2.00283 


2.00418 


2.00553 


2.00687 


2.00821 


2.00956 


2.01089 


2.01223 


2.01357 


7.5 


2.01490 


2.01624 


2.01757 


2.01890 


2.02022 


2.02155 


2.02287 


2.02419 


2.02551 


2.02683 


7.6 


2.02815 


2.02946 


2.03078 


2.03209 


2.03340 


2.03471 


2.03601 


2.03732 


2.03862 


2.03992 


7.7 


2.04122 


2.04252 


2.04381 


2.04511 


2.04640 


2.04769 


2.04898 


2.05027 


2.06156 


2.05284 


7.8 


2.05412 


2.05540 


2.05668 


2.05796 


2.05924 


2.06051 


2.06179 


2.06306 


2.06433 


2.06560 


7.9 


2.06686 


2.06813 


2.06939 


2.07065 


2.07191 


2.07317 


2.07443 


2.07568 


2.07694 


2.07819 


8.0 


2.07944 


2.08069 


2.08194 


2.08318 


2.08443 


2.08567 


2.08691 


2.08815 


2.08939 


2.09063 


8.1 


2.09186 


2.09310 


2.09433 


2.09556 


2.09679 


2.09802 


2.09924 


2.10047 


2.10169 


2.10291 


8.2 


2.10413 


2.10535 


2.10657 


2.10779 


2.109O0 


2.11021 


2.11142 


2.11263 


2.11384 


2.11505 


8.3 


2.11626 


2.11746 


2.11866 


2.11986 


2.12106 


2.12226 


2.12346 


2.12466 


2.12585 


2.12704 


8.4 


2.12823 


2.12942 


2.13061 


2.13180 


2.13298 


2.13417 


2.13535 


2.13653 


2.13771 


2.13889 


8.5 


2.14007 


2.14124 


2.14242 


2.14359 


2.14476 


2.14593 


2.14710 


2.14827 


2.14943 


2.15060 


8.6 


2.15176 


2.15292 


2.15409 


2.15524 


2.15640 


2.15756 


2.15871 


2.15987 


2.16102 


2.16217 


8.7 


2.16332 


2.16447 


2.16562 


2.16677 


2.16791 


2.16906 


2.17020 


2.17134 


2.17248 


2.17361 


8.8 


2.17475 


2.17589 


2.17702 


2.17816 


2.17929 


2.18042 


2.18155 


2.18267 


2.18380 


2.18493 


8.9 


2.18605 


2.18717 


2.18830 


2.18942 


2.19054 


2.19165 


2.19277 


2.19389 


2.19500 


2.19611 


9.0 


2.19722 


2.19834 


2.19944 


2.20055 


2.20166 


2.20276 


2.20387 


2.20497 


2.20607 


2.20717 


9.1 


2.20827 


2.20937 


2.21047 


2.21157 


2.21266 


2.21375 


2.21485 


2.21594 


2.21703 


2.21812 


9.2 


2.21920 


2.22029 


2.22138 


2.22246 


2.22354 


2.22462 


2.22570 


2.22678 


2.22786 


2.22894 


9.3 


2.23001 


2.23109 


2.23216 


2.23324 


2.23431 


2.23538 


2.23645 


2.23761 


2.23858 


2.23965 


9.4 


2'.24071 


2.24177 


2.24284 


2.24390 


2.24496 


2.24601 


2.24707 


2.24813 


2.24918 


2.25024 


9.5 


2.25129 


2.25234 


2.25339 


2.25444 


2.25549 


2.25654 


2.25759 


2.25863 


2.25968 


2.26072 


9.6 


2.26176 


2.26280 


2.26384 


2.26488 


2.26592 


2.26696 


2.26799 


2.26903 


2.27006 


2.27109 


9.7 


2.27213 


2.27316 


2.27419 


2.27521 


2.27624 


2.27727 


2.27829 


2.27932 


2.28034 


2.28136 


9.8 


2.28238 


2.28340 


2.28442 


2.28544 


2.28646 


2.28747 


2.28849 


2.28950 


2.29051 


2.29152 


9.9 


2.29253 


2.29354 


2.29455 


2.29556 


2.29657 


2.29757 


2.29858 


2.29958 


2.30068 


2.30158 



225 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


.0 


1.0000 


1.0101 


1.0202 


1.0305 


1.0408 


1.0513 


1.0618 


1.0725 


1.0833 


1.0942 


.1 


1.1052 


1.1163 


1.1275 


1.1388 


1.1503 


1.1618 


1.1735 


1.1853 


1.1972 


1.2092 


.2 


1.2214 


1.2337 


1.2461 


1.2586 


1.2712 


1.2840 


1.2969 


1.3100 


1.3231 


1.3364 


.3 


1.3499 


1.3634 


1.3771 


1.3910 


1.4049 


1.4191 


1.4333 


1.4477 


1.4623 


1.4770 


.4 


1.4918 


1.5068 


1.5220 


1.5373 


1.5527 


1.5683 


1.5841 


1.6000 


1.6161 


1.6323 


.5 


1.6487 


1.6653 


1.6820 


1.6989 


1.7160 


1.7333 


1.7507 


1.7683 


1.7860 


1.8040 


.6 


1.8221 


1.8404 


1.8589 


1.8776 


1.8965 


1.9155 


1.9348 


1.9542 


1.9739 


1.9937 


.7 


2.0138 


2.0340 


2.0544 


2.0751 


2.0959 


2.1170 


2.1383 


2.1598 


2.1815 


2.2034 


.8 


2.2255 


2.2479 


2.2705 


2.2933 


2.3164 


2.3396 


2.3632 


2.38G9 


2.4109 


2.4351 


.9 


2.4596 


2.4843 


2.5093 


2.5345 


2.5600 


2.5857 


2,6117 


2.6379 


2.6645 


2.6912 


1.0 


" 2.7183 


2.7456 


2.7732 


2.8011 


2.8292 


2.8577 


2.8864 


2,9154 


2.9447 


2.9743 


1.1 


3.0042 


3.0344 


3.0649 


3.0957 


3.1268 


3.1582 


3.1899 


3.2220 


3.2544 


3.2871 


1.2 


3.3201 


3.3535 


3.3872 


3.4212 


3.4556 


3.4903 


3.5254 


3.5609 


3.5966 


3.6328 


1.3 


3.6693 


3.7062 


3.7434 


3.7810 


3.8190 


3.8574 


3.8962 


3.9354 


3.9749 


4.0149 


1.4 


4.0552 


4.0960 


4.1371 


4.1787 


4.2207 


4.2631 


4.3060 


4.3492 


4.3929 


4.4371 


1.5 


4.4817 


4.5267 


4.5722 


4.6182 


4.6646 


4.7115 


4.7588 


4.8066 


4.8550 


4.9037 


1.6 


4.9530 


5.0028 


5.0531 


5.1039 


5.1552 


5.2070 


5.2593 


5.3122 


5.3656 


5.4195 


1.7 


5.4739 


5.5290 


5.5845 


5.6407 


5.6973 


5.7546 


5.8124 


5.8709 


5.9299 


5.9895 


1.8 


6.0496 


6.1104 


6.1719 


6.2339 


6.2965 


6.3598 


6.4237 


6.4883 


6.5535 


6.6194 


1.9 


6.6859 


6.7531 


6.8210 


6.8895 


6.9588 


7.0287 


7.0993 


7.1707 


7.2427 


7.3155 


2.0 


7.3891 


7.4633 


7.5383 


7.6141 


7.6906 


7.7679 


7.8460 


7.9248 


8.0045 


8.0849 


2.1 


8.1662 


8.2482 


8.3311 


8.4149 


8.4994 


8.5849 


8.6711 


8.7583 


8.8463 


8.9352 


2.2 


9.0250 


9.1157 


9.2073 


9.2999 


9.3933 


9.4877 


9.5831 


9.G794 


9.7767 


9.8749 


2.3 


9.9742 


10.074 


10.176 


10.278 


10.381 


10.486 


10.591 


10.697 


10.805 


10.913 


2.4 


11.023 


11.134 


11.246 


11.359 


11.473 


11.588 


11.705 


11.822 


11.941 


12.061 


2.5 


12.182 


12.305 


12.429 


12.554 


12.680 


12.807 


12.936 


13.066 


13.197 


13.330 


2.6 


13.464 


13.599 


13.73G 


13.874 


14.013 


14.154 


14.296 


14.440 


14.585 


14.732 


2.7 


14.880 


15.029 


15.180 


15.333 


15.487 


15.643 


15.800 


15.959 


16.119 


16.281 


2.8 


16.445 


16.610 


16.777 


16.945 


17.116 


17.288 


17.462 


17.637 


17.814 


17.993 


2.9 


18.174 


18.357 


18.541 


18.728 


18.916 


19.106 


19.298 


19.492 


19.688 


19.886 


3.0 


20.086 


20.287 


20.491 


20.697 


20.905 


21.115 


21.328 


21.542 


21.758 


21.977 


3.1 


22.198 


22.421 


22.646 


22.874 


23.104 


23.336 


23.571 


23.807 


24.047 


24.288 


3.2 


24.533 


24.779 


25.028 


25.280 


25.534 


25.790 


26.050 


26.311 


26.576 


26.843 


3.3 


27.113 


27.385 


27.660 


27.938 


28.219 


28.503 


28.789 


29.079 


29.371 


29.666 


3.4 


29.964 


30.265 


30.569 


30.877 


31.187 


31.500 


31.817 


32.137 


32.460 


32.786 


3.5 


33.115 


33.448 


33.784 


34.124 


34.467 


34.813 


35.163 


35.517 


35.874 


36.234 


3.6 


36.598 


36.966 


37.338 


37.713 


38.092 


38.475 


38.861 


39.252 


39.646 


40.045 


3.7 


40.447 


40.854 


41.264 


41.679 


42.098 


42.521 


42.948 


43.380 


43.816 


44.256 


3.8 


44.701 


45.150 


45.604 


46.063 


46.525 


46.993 


47.465 


47.942 


48.424 


48.911 


3.9 


49.402 


49.899 


50.400 


50.907 


51.419 


51.935 


52.457 


52.985 


53.517 


54.055 


4. 


54.598 


60.340 


66.686 


73.700 


81.451 


90.017 


99.484 


109.95 


121.51 


134.29 


5. 


148.41 


164.02 


181.27 


200.34 


221.41 


244.69 


270.43 


298.87 


330.30 


365.04 


6. 


403.43 


445.86 


492.75 


544.57 


601.85 


665.14 


735.10 


812.41 


897.85 


992.27 


7. 


1096.6 


1212.0 


1339.4 


1480.3 


1636.0 


1808.0 


1998.2 


2208.3 


2440.6 


2697.3 


8. 


2981.0 


3294.5 


3641.0 


4023.9 


4447.1 


4914.8 


5431.7 


6002.9 


6634.2 


7332.0 


9. 


8103.1 


8955.3 


9897.1 


10938 


12088 


13360 


14765 


16318 


18034 


19930 


10. 


22026 





















226 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


.0 


1.00000 


.99005 


.98020 


.97045 


.96079 


.95123 


.94176 


.93239 


.92312 


.91393 


.1 


.90484 


.89583 


.88692 


.87810 


.86936 


.86071 


.85214 


.84366 


.83527 


.82696 


.2 


.81873 


.81058 


.80252 


.79453 


.78663 


.77880 


.77105 


.76338 


.75578 


.74826 


.3 


.74082 


.73345 


.72615 


.71892 


.71177 


.70469 


.69768 


.69073 


.68386 


.67706 


.4 


.67032 


.66365 


.65705 


.65051 


.64404 


.63763 


.63128 


.62500 


.61878 


.61263 


.5 


.60653 


.60050 


.59452 


.58860 


.58275 


.57695 


.57121 


.56553 


.55990 


.55433 


.6 


.54881 


.54335 


.53794 


.53259 


.52729 


.52205 


.51685 


.51171 


.50662 


.50158 


.7 


.49659 


.49164 


.48675 


.48191 


.47711 


.47237 


.46767 


.46301 


.45841 


.45384 


.8 


.44933 


.44486 


.44043 


.43605 


.43171 


.42741 


.42316 


.41895 


.41478 


.41066 


.9 


.40657 


.40252 


.39852 


.39455 


.39063 


.38674 


.38289 


.37908 


.37531 


.37158 


1.0 


.36788 


.36422 


.36060 


.35701 


.35345 


.34994 


.34646 


.34301 


.33960 


.33622 


1.1 


.33287 


.32956 


.32628 


.32303 


.31982 


.31664 


.31349 


.31037 


.30728 


.30422 


1.2 


.30119 


.29820 


.29523 


.29229 


.28938 


.28650 


.28365 


.28083 


.27804 


.27527 


1.3 


.27253 


.26982 


.26714 


.26448 


.26185 


.25924 


.25666 


.25411 


.25158 


.24908 


1.4 


.24660 


.24414 


.24171 


.23931 


.23693 


.23457 


.23224 


.22993 


.22764 


.22537 


1.5 


.22313 


.22091 


.21871 


.21654 


.21438 


.21225 


.21014 


.20805 


.20598 


.20393 


1.6 


.20190 


.19989 


.19790 


.19593 


.19398 


.19205 


.19014 


.18825 


.18637 


.18452 


1.7 


.18268 


.18087 


.17907 


.17728 


.17552 


.17377 


.17204 


.17033 


.16864 


.16696 


1.8 


.16530 


.16365 


.16203 


.16041 


.15882 


.15724 


.15567 


.15412 


.15259 


.15107 


1.9 


.14957 


.14808 


.14661 


.14515 


.14370 


.14227 


.14086 


.13946 


.13807 


.13670 


2.0 


.13534 


.13399 


.13266 


.13134 


.13003 


.12873 


.12745 


.12619 


.12493 


.12369 


2.1 


.12246 


.12124 


.12003 


.11884 


.11765 


.11648 


.11533 


.11418 


.11304 


.11192 


2.2 


.11080 


.10970 


.10861 


.10753 


.10646 


.10540 


.10435 


.10331 


.10228 


.10127 


2.3 


.10026 


.09926 


.09827 


.09730 


.09633 


.09537 


.09442 


.09348 


.09255 


.09163 


2.4 


.09072 


.08982 


.08892 


.08804 


.08716 


.08629 


.08543 


.08458 


.08374 


.08291 


2.5 


.08208 


.08127 


.08046 


.07966 


.07887 


.07808 


.07730 


.07654 


.07577 


.07502 


2.6 


.07427 


.07363 


.07280 


.07208 


.07136 


.07065 


.06995 


.06925 


.06856 


.06788 


2.7 


.06721 


.06654 


.06587 


.06522 


.06457 


.06393 


.06329 


.06266 


.06204 


.06142 


2.8 


.06081 


.06020 


.05961 


.05901 


.05843 


.05784 


.05727 


.05670 


.05613 


.05558 


2.9 


.05502 


.05448 


.05393 


.05340 


.05287 


.05234 


.05182 


.05130 


.05079 


.05029 


3.0 


.04979 


.04929 


.04880 


.04832 


.04783 


.04736 


.04689 


.04642 


.04596 


.04550 


3.1 


.04505 


.04460 


.04416 


.04372 


.04328 


.04285 


.04243 


.04200 


.04159 


.04117 


3.2 


.04076 


.04036 


.03996 


.03956 


.03916 


.03877 


.03839 


.03801 


.03763 


.03725 


3.3 


.03688 


.03652 


.03615 


.03579 


.03544 


.03508 


.03474 


.03439 


.03405 


.03371 


3.4 


.03337 


.03304 


.03271 


.03239 


.03206 


.03175 


.03143 


.03112 


.03081 


.03050 


3.5 


.03020 


.02990 


.02960 


.02930 


.02901 


.02872 


.02844 


.02816 


.02788 


.02760 


3.6 


.02732 


.02705 


.02678 


.02652 


.02625 


.02599 


.02573 


.02548 


.02522 


.02497 


3.7 


.02472 


.02448 


.02423 


.02399 


.02375 


.02352 


.02328 


.02305 


.02282 


.02260 


3.8 


.02237 


.02215 


.02193 


.02171 


.02149 


.02128 


.02107 


.02086 


.02065 


.02045 


3.9 


.02024 


.02004 


.01984 


.01964 


.01945 


.01925 


.01906 


.01887 


.01869 


.01850 


4. 


.018316 


.016573 


.014996 


.013569 


.012277 


.011109 


.010052 


.0290953 


.0282297 


.0274466 


5. 


.0267379 


.0260967 


.0255166 


.0249916 


.0245166 


.0240868 


.0236979 


.0233460 


.0230276 


.0227394 


6. 


.0224788 


.0222429 


.0220294 


.0218363 


.0216616 


.0215034 


.0213604 


.0212309 


.0211138 


.0210078 


7. 


.0^91188 


.0382510 


.0374659 


.0367554 


.0^61125 


.0355308 


.0350045 


.0345283 


.0340973 


.0337074 


8. 


.0333546 


.0^30354 


.0327465 


.0^24852 


.0322487 


.0320347 


.0318411 


.0316659 


.0315073 


.0313639 


9. 


.0^12341 


.0^11167 


.0310104 


.0'91424 


.0''82724 


.0-'74852 


.0''67729 


.O'' 61 283 


.0^55452 


.0^50175 


10. 


.0^45400 


- C.OOOQlt^'i^') 

















227 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


.0 


.0000 


.0100 


.0200 


.0300 


.0400 


.0500 


.0600 


.0701 


.0801 


.0901 


.1 


.1002 


.1102 


.1203 


.1304 


.1405 


.1506 


.1607 


.1708 


.1810 


.1911 


.2 


.2013 


.2115 


.2218 


.2320 


.2423 


.2526 


.2629 


.2733 


.2837 


.2941 


.3 


.3045 


.3150 


.3255 


.3360 


.3466 


.3572 


.3678 


.3785 


.3892 


.4000 


.4 


.4108 


.4216 


.4325 


.4434 


.4543 


.4653 


.4764 


.4875 


.4986 


.5098 


.5 


.5211 


.5324 


.5438 


.5552 


.5666 


.5782 


.5897 


.6014 


.6131 


.6248 


.6 


.6367 


.6485 


.6605 


.6725 


.6846 


.6967 


.7090 


.7213 


.7336 


.7461 


.7 


.7586 


.7712 


.7838 


.7966 


.8094 


.8223 


.8353 


.8484 


.8615 


.8748 


.8 


.8881 


.9015 


.9150 


.9286 


.9423 


.9561 


.9700 


.9840 


.9981 


1.0122 


.9 


1.0265 


1.0409 


1.0554 


1.0700 


1.0847 


1.0995 


1.1144 


1.1294 


1.1446 


1.1598 


1.0 


1.1752 


1.1907 


1.2063 


1.2220 


1.2379 


1.2539 


1.2700 


1.2862 


1.3025 


1.3190 


1.1 


1.3356 


1.3524 


1.3693 


1.3863 


1.4035 


1.4208 


1.4382 


1.4558 


1.4735 


.1.4914 


1.2 


1.5095 


1.5276 


1.5460 


1.5645 


1.5831 


1.6019 


1.6209 


1.6400 


1.6593 


1.6788 


1.3 


1.6984 


1.7182 


1.7381 


1.7583 


1.7786 


1.7991 


1.8198 


1.8406 


1.8617 


1.8829 


1.4 


1.9043 


1.9259 


1.9477 


1.9697 


1.9919 


2.0143 


2.0369 


2.0597 


2.0827 


2.1059 


1.5 


2.1293 


2.1529 


2.1768 


2.2008 


2.2251 


2.2496 


2.2743 


2.2993 


2.3245 


2.3499 


1.6 


2.3756 


2.4015 


2.4276 


2.4540 


2.4806 


2.5075 


2.5346 


2.5620 


2.5896 


2.6175 


1.7 


2.6456 


2.6740 


2.7027 


2.7317 


2.7609 


2.7904 


2.8202 


2.8503 


2.8806 


2.9112 


1.8 


2.9422 


2.9734 


3.0049 


3.0367 


3.0689 


3.1013 


3.1340 


3.1671 


3.2005 


3.2341 


1.9 


3.2682 


3.3025 


3.3372 


3.3722 


3.4075 


3.4432 


3.4792 


3.5156 


3.5523 


3.5894 


2.0 


3.6269 


3.6647 


3.7028 


3.7414 


3.7803 


3.8196 


3.8593 


3.8993 


3.9398 


3.9806 


2.1 


4.0219 


4.0635 


4.1056 


4.1480 


4.1909 


4.2342 


4.2779 


4.3221 


4.3666 


4.4116 


2.2 


4.4571 


4.5030 


4.5494 


4.5962 


4.6434 


4.6912 


4.7394 


4.7880 


4.8372 


4.8868 


2.3 


4.9370 


4.9876 


5.0387 


5.0903 


5.1425 


5.1951 


5.2483 


5.3020 


5.3562 


5.4109 


2.4 


5.4662 


5.5221 


5.5785 


5.6354 


5.6929 


5.7510 


5.8097 


5.8689 


5.9288 


5.9892 


2.5 


6,0502 


6.1118 


6.1741 


6.2369 


6.3004 


6.3645 


6.4293 


6.4946 


6.5607 


6.6274 


2.6 


6.6947 


6.7628 


6.8315 


6.9008 


6.9709 


7.0417 


7.1132 


7.1854 


7.2583 


7.3319 


2.7 


7.4063 


7.4814 


7.5572 


7.6338 


7.7112 


7.7894 


7.8683 


7.9480 


8.0285 


8.1098 


2.8 


8.1919 


8.2749 


8.3586 


8.4432 


8.5287 


8.6150 


8.7021 


8.7902 


8.8791 


8.9689 


2.9 


9.0596 


9.1512 


9.2437 


9.3371 


9.4315 


9.5268 


9.6231 


9.7203 


9.8185 


9.9177 



228 



Table 18a 

(continued) 



HYPERBOLIC FUNCTIONS 

sinh X 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


3.0 


10.018 


10.119 


10.221 


10.324 


10.429 


10.534 


10.640 


10.748 


10.856 


10.966 


3.1 


11.076 


11.188 


11.301 


11.415 


11.530 


11.647 


11.764 


11.883 


12.003 


12.124 


3.2 


12.246 


12.369 


12.494 


12.620 


12.747 


12.876 


13.006 


13.137 


13.269 


13.403 


3.3 


13.538 


13.674 


13.812 


13.951 


14.092 


14.234 


14.377 


14.522 


14.668 


14.816 


3.4 


14.965 


15.116 


15.268 


15.422 


15.577 


15.734 


15.893 


16.053 


16.215 


16.378 


3.5 


16.543 


16.709 


16.877 


17.047 


17.219 


17.392 


17.567 


17.744 


17.923 


18.103 


3.6 


18.285 


18.470 


18.655 


18.843 


19.033 


19.224 


19.418 


19.613 


19.811 


20.010 


3.7 


20.211 


20.415 


20.620 


20.828 


21.037 


21.249 


21.463 


21.679 


21.897 


22.117 


3.8 


22.339 


22.564 


22.791 


23.020 


23.252 


23.486 


23.722 


23.961 


24.202 


24.445 


3.9 


24.691 


24.939 


25.190 


25.444 


25.700 


25.958 


26.219 


26.483 


26.749 


27.018 


4.0 


27.290 


27.564 


27.842 


28.122 


28.404 


28.690 


28.979 


29.270 


29.564 


29.862 


4.1 


30.162 


30.465 


30.772 


31.081 


31.393 


31.709 


32.028 


32.350 


32.675 


33.004 


4.2 


33.336 


33.671 


34.009 


34.351 


34.697 


35.046 


35.398 


35.754 


36.113 


36.476 


4.3 


36.843 


37.214 


37.588 


37.965 


38.347 


38.733 


39.122 


39.515 


39.913 


40.314 


4.4 


40.719 


41.129 


41.542 


41.960 


42.382 


42.808 


43.238 


43.673 


44.112 


44.555 


4.5 


45.003 


45.455 


45.912 


46.374 


46.840 


47.311 


47.787 


48.267 


48.752 


49.242 


4.6 


49.737 


60.237 


50.742 


51.252 


51.767 


52.288 


52.813 


63.344 


53.880 


54.422 


4.7 


54.969 


55.522 


56.080 


56.643 


57.213 


57.788 


58.369 


58.995 


59.548 


60.147 


4.8 


60.751 


61.362 


61.979 


62.601 


63.231 


63.866 


64.508 


65.157 


65.812 


66.473 


4.9 


67.141 


67.816 


68.498 


69.186 


69.882 


70.584 


71.293 


72.010 


72.734 


73.465 


5.0 


74.203 


74.949 


75.702 


76.463 


77.232 


78.008 


78.792 


79.584 


80.384 


81.192 


5.x 


82.008 


82.832 


83.665 


84.506 


85.355 


86.213 


87.079 


87.955 


88.839 


89.732 


5.2 


90.633 


91.544 


92.464 


93.394 


94.332 


95.281 


96.238 


97.205 


98.182 


99.169 


5.3 


100.17 


101.17 


102.19 


103.22 


104.25 


105.30 


106.36 


107.43 


108.51 


109.60 


5.4 


110.70 


111.81 


112.94 


114.07 


115.22 


116.38 


117.55 


118.73 


119.92 


121.13 


5.5 


122.34 


123.57 


124.82 


126.07 


127.34 


128.62 


129.91 


131.22 


132.53 


133.87 


5.6 


135.21 


136.57 


137.94 


139.33 


140.73 


142.14 


143.57 


145.02 


146.47 


147.95 


5.7 


149.43 


150.93 


152.45 


153.98 


155.53 


157.09 


158.67 


160.27 


161.88 


163.51 


5.8 


165.15 


166.81 


168.48 


170.18 


171.89 


173.62 


175.36 


177.12 


178.90 


180.70 


5.9 


182.52 


184.35 


186.20 


188.08 


189.97 


191.88 


193.80 


195.75 


197.72 


199.71 



229 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


.0 


1.0000 


1.0001 


1.0002 


1.0005 


1.0008 


1.0013 


1.0018 


1.0025 


1.0032 


1.0041 


.1 


1.0050 


1.0061 


1.0072 


1.0085 


1.0098 


1.0113 


1.0128 


1.0145 


1.0162 


1.0181 


.2 


1.0201 


1.0221 


1.0243 


1.0266 


1.0289 


1.0314 


1.0340 


1.0367 


1.0395 


1.0423 


.3 


1.0453 


1.0484 


1.0516 


1.0549 


1.0584 


1.0619 


1.0655 


1,0692 


1.0731 


1.0770 


A 


1.0811 


1.0852 


1.0895 


1.0939 


1.0984 


1.1030 


1.1077 


1,1125 


1.1174 


1.1225 


.5 


1.1276 


1.1329 


1.1383 


1.1438 


1,1494 


1.1551 


1.1609 


1.1669 


1.1730 


1.1792 


.6 


1.1855 


1.1919 


1.1984 


1.2051 


1.2119 


1.2188 


1.2258 


1.2330 


1.2402 


1.2476 


.7 


1.2552 


1.2628 


1.2706 


1.2785 


1.2865 


1.2947 


1.3030 


1.3114 


1.3199 


1.3286 


.8 


1.3374 


1.3464 


1.3555 


1.3647 


1.3740 


1.3835 


1.3932 


1.4029 


1.4128 


1.4229 


.9 


1.4331 


1.4434 


1.4539 


1.4645 


1.4753 


1.4862 


1.4973 


1.5085 


1.5199 


1.5314 


1.0 


1.5431 


1.5549 


1.5669 


1.5790 


1.5913 


1.6038 


1.6164 


1.6292 


1.6421 


1.6552 


1.1 


1.6685 


1.6820 


1.6956 


1.7093 


1.7233 


1.7374 


1.7517 


1.7662 


1.7808 


1.7957 


1.2 


1.8107 


1.S258 


1.8412 


1.8568 


1.8725 


1.8884 


1.9045 


1.9208 


1.9373 


1.9540 


1.3 


1.9709 


1.9880 


2.0053 


2.0228 


2.0404 


2.0583 


2.0764 


2.0947 


2.1132 


2.1320 


1.4 


2.1509 


2.1700 


2.1894 


2.2090 


2.2288 


2.2488 


2.2691 


2.2896 


2.3103 


2.3312 


1.5 


2.3524 


2.3738 


2.3955 


2.4174 


2.4395 


2.4619 


2.4845 


2.5073 


2.5305 


2.5538 


1.6 


2.5775 


2.6013 


2.6255 


2.6499 


2.6746 


2.6995 


2.7247 


2.7502 


2.7760 


2.8020 


1.7 


2.8283 


2.8549 


2.8818 


2.9090 


2.9364 


2.9642 


2.9922 


3.0206 


3.0492 


3.0782 


1.8 


3.1075 


3.1371 


3.1669 


3.1972 


3.2277 


3.2585 


3.2897 


3.3212 


3.3530 


3.3852 


1.9 


3.4177 


3.4506 


3.4838 


3.5173 


3.5512 


3.5855 


3.6201 


3.6551 


3.6904 


3.7261 


2.0 


3.7622 


3.7987 


3.8355 


3.8727 


3.9103 


3.9483 


3.9867 


4,0255 


4.0647 


4.1043 


2.1 


4.1443 


4.1847 


4.2256 


4.2669 


4.3085 


4.3507 


4.3932 


4.4362 


4.4797 


4.5236 


2.2 


4.5679 


4.6127 


4.6580 


4.7037 


4.7499 


4.7966 


4.8437 


4.8914 


4.9395 


4.9881 


2.3 


5.0372 


5.0868 


5.1370 


5.1876 


5.2388 


5.2905 


5.3427 


5.3954 


5.4487 


5.5026 


2.4 


5.5569 


5.6119 


5.6674 


6.7235 


5.7801 


5.8373 


5.8951 


5,9535 


6.0125 


6.0721 


2.5 


6.1323 


6.1931 


6.2545 


6.3166 


6.3793 


6.4426 


6.5066 


6.5712 


6.6365 


6.7024 


2.6 


6.7690 


6.8363 


6.9043 


6.9729 


7.0423 


7.1123 


7.1831 


7.2546 


7.3268 


7.3998 


2.7 


7.4735 


7.5479 


7.6231 


7.6991 


7.7758 


7.8533 


7.9316 


8.0106 


8.0905 


8.1712 


2.8 


8.2527 


8.3351 


8.4182 


8.5022 


8.5871 


8.6728 


8.7594 


8.8469 


8.9352 


9.0244 


2.9 


9.1146 


9.2056 


9.2976 


9.3905 


9.4844 


9.5791 


9.6749 


9.7716 


9.8693 


9.9680 



230 



Table 18b 

(con tinned) 



HYPERBOLIC FUNCTIONS 



cosh X 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


3.0 


10.068 


10.168 


10.270 


10.373 


10.476 


10.581 


10.687 


10.794 


10.902 


11.011 


3.1 


11.121 


11.233 


11.345 


11.459 


11.574 


11.689 


11.806 


11.925 


12.044 


12.165 


3.2 


12.287 


12.410 


12.534 


12.660 


12.786 


12.915 


13.044 


13.175 


13.307 


13.440 


3.3 


13.575 


13.711 


13.848 


13.987 


14.127 


14.269 


14.412 


14.556 


14.702 


14.850 


3.4 


14.999 


15.149 


15.301 


15.455 


15.610 


15.766 


15.924 


16.084 


16.245 


16.408 


3.5 


16.573 


16.739 


16.907 


17.077 


17.248 


17.421 


17.596 


17.772 


17.951 


18.131 


3.6 


18.313 


18.497 


18.682 


18.870 


19.059 


19.250 


19.444 


19.639 


19.836 


20.035 


3.7 


20.236 


20.439 


20.644 


20.852 


21.061 


21.272 


21.486 


21.702 


21.919 


22.139 


3.8 


22.302 


22.586 


22.813 


23.042 


23.273 


23.507 


23.743 


23.982 


24.222 


24.466 


3.9 


24.711 


24.959 


25.210 


25.463 


25.719 


25.977 


26.238 


26.502 


26.768 


27.037 


4.0 


27.308 


27.583 


27.860 


28.139 


28.422 


28.707 


28.996 


29.287 


29.581 


29.878 


4.1 


30.178 


30.482 


30.788 


31.097 


31.409 


31.725 


32.044 


32.365 


32.691 


33.019 


4.2 


33.351 


33.686 


34.024 


34.366 


34.711 


35.060 


35.412 


35.768 


36.127 


36.490 


4.3 


36.857 


37.227 


37.601 


37.979 


38.360 


38.746 


39.135 


39.528 


39.925 


40.326 


4.4 


40.732 


41.141 


41.554 


41.972 


42.393 


42.819 


43.250 


43.684 


44.123 


44.566 


4.5 


45.014 


45.466 


45.923 


46.385 


46.851 


47.321 


47.797 


48.277 


48.762 


49.252 


4.6 


49.747 


50.247 


50.752 


51.262 


51.777 


52.297 


52.823 


53.354 


53.890 


54.431 


4.7 


54.978 


55.531 


66.089 


56.652 


57.221 


57.796 


58.377 


58.964 


59.556 


60.155 


4.8 


60.759 


61.370 


61.987 


62.609 


63.239 


63.874 


64.516 


65.164 


65.819 


66.481 


4.9 


67.149 


67.823 


68.505 


69.193 


69.889 


70.591 


71.300 


72.017 


72.741 


73.472 


5.0 


74.210 


74.956 


75.709 


76.470 


77.238 


78.014 


78.798 


79.590 


80.390 


81.198 


5.1 


82.014 


82.838 


83.671 


84.512 


85.361 


86.219 


87.085 


87.960 


88.844 


89.737 


5.2 


90.639 


91.550 


92.470 


93.399 


94.338 


95.286 


96.243 


97.211 


98.188 


99.174 


5.3 


100.17 


101.18 


102.19 


103.22 


104.26 


105.31 


106.67 


107.43 


108.51 


109.60 


5.4 


110.71 


111.82 


112.94 


114.08 


115.22 


116.38 


117.55 


118.73 


119.93 


121.13 


5.5 


122.35 


123.58 


124.82 


126.07 


127.34 


128.62 


129.91 


131.22 


132.54 


133.87 


5.6 


135.22 


136.57 


137.95 


139.33 


140.73 


142.15 


143.58 


145.02 


146.48 


147.95 


5.7 


149.44 


150.94 


152.45 


153.99 


155.53 


157.10 


158.68 


160.27 


161.88 


163.51 


5.8 


165.15 


166.81 


168.49 


170.18 


171.89 


173.62 


175.36 


177.13 


178.91 


180.70 


5.9 


182.52 


184.35 


186.21 


188.08 


189.97 


191.88 


193.81 


195.75 


197.72 


199.71 



231 




X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


.0 


.00000 


.01000 


.02000 


.02999 


.03998 


.04996 


.05993 


.06989 


.07983 


.08976 


.1 


.09967 


.10956 


.11943 


.12927 


.13909 


.14889 


.15865 


.16838 


.17808 


.18775 


.2 


.19738 


.20697 


.21652 


.22603 


.23550 


.24492 


.25430 


.26362 


.27291 


.28213 


.3 


.29131 


.30044 


.30951 


.31852 


.32748 


.33638 


.34521 


.35399 


.36271 


.37136 


.4 


.37995 


.38847 


.39693 


.40532 


.41364 


.42190 


.43008 


.43820 


.44624 


.45422 


.5 


.46212 


.46995 


.47770 


.48538 


.49299 


.50052 


.50798 


.51536 


.52267 


.52990 


.6 


.53705 


.54413 


.55113 


.55805 


.56490 


.57167 


.57836 


.58498 


.59152 


.59798 


.7 


.60437 


.61068 


.61691 


.62307 


.62915 


.63515 


.64108 


.64693 


.65271 


.65841 


.8 


.66404 


.66959 


.67507 


.68048 


.68581 


.69107 


.69626 


.70137 


.70642 


.71139 


.9 


.71630 


.72113 


.72590 


.73059 


.73522 


.73978 


.74428 


.74870 


.75307 


.75736 


1.0 


.76159 


.76576 


.76987 


.77391 


.77789 


.78181 


.78566 


.78946 


.79320 


.79688 


1.1 


.80050 


.80406 


.80757 


.81102 


.81441 


.81775 


.82104 


.82427 


.82745 


.83058 


1.2 


.83365 


.83668 


.83965 


.84258 


.84546 


.84828 


.85106 


.85380 


.85648 


.85913 


1.3 


.86172 


.86428 


.86678 


.86925 


.87167 


.87405 


.87639 


.87869 


.88095 


.88317 


1.4 


.88535 


.88749 


.88960 


.89167 


.89370 


.89569 


.89765 


.89958 


.90147 


.90332 


1.5 


.90515 


.90694 


.90870 


.91042 


.91212 


.91379 


.91542 


.91703 


.91860 


.92015 


1.6 


.92167 


.92316 


.92462 


.92606 


.92747 


.92886 


.93022 


.93155 


.93286 


.93415 


1.7 


.93541 


.93665 


.93786 


.93906 


.94023 


.94138 


.94250 


.94361 


.94470 


.94576 


1.8 


.94681 


.94783 


.94884 


.94983 


.95080 


.95175 


.95268 


.95359 


.95449 


.95537 


1.9 


.95624 


.95709 


.95792 


.95873 


.95953 


.96032 


.96109 


.96185 


.96259 


.96331 


2.0 


.96403 


.96473 


.96541 


.96609 


.96675 


.96740 


.96803 


.96865 


.96926 


.96986 


2.1 


.97045 


.97103 


.97159 


.97215 


.97269 


.97323 


.97375 


.97426 


.97477 


.97526 


2.2 


.97574 


.97622 


.97668 


.97714 


.97759 


.97803 


.97846 


.97888 


.97929 


.97970 


2.3 


.98010 


.98049 


.98087 


.98124 


.98161 


.98197 


.98233 


.98267 


.98301 


.98335 


2.4 


.98367 


.98400 


.98431 


.98462 


.98492 


.98522 


.98551 


.98579 


.98607 


.98635 


2.5 


.98661 


.98688 


.98714 


.98739 


.98764 


.98788 


.98812 


.98835 


.98858 


.98881 


2.6 


.98903 


.98924 


.98946 


.98966 


.98987 


.99007 


.99026 


.99045 


.99064 


.99083 


2.7 


.99101 


.99118 


.99136 


.99153 


.99170 


.99186 


.99202 


.99218 


.99233 


.99248 


2.8 


.99263 


.99278 


.99292 


.99306 


.99320 


.99333 


.99346 


.99359 


.99372 


.99384 


2.9 


.99396 


.99408 


.99420 


.99431 


.99443 


.99454 


.99464 


.99475 


.99485 


.99496 



232 



Table 18c 

(continued) 



HYPERBOLIC FUNCTIONS 

tanh X 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


3.0 


.99505 


.99515 


.99525 


.99534 


.99543 


.99552 


.99561 


.99570 


.99578 


.99587 


3.1 


.99595 


.99603 


.99611 


.99618 


.99626 


.99633 


.99641 


.99648 


.99655 


.99662 


3.2 


.99668 


.99675 


.99681 


.99688 


.99694 


.99700 


.99706 


.99712 


.99717 


.99723 


3.3 


.99728 


.99734 


.99739 


.99744 


.99749 


.99754 


.99759 


.99764 


.99768 


.99773 


3.4 


.99777 


.99782 


.99786 


.99790 


.99795 


.99799 


.99803 


.99807 


.99810 


.99814 


3.5 


.99818 


.99821 


.99825 


.99828 


.99832 


.99835 


.99838 


.99842 


.99845 


.99848 


3.6 


.99851 


.99853 


.99857 


.99859 


.99862 


.99865 


.99868 


.99870 


.99873 


.99875 


3.7 


.99878 


.99880 


.99883 


.99885 


.99887 


.99889 


.99892 


.99894 


.99896 


.99898 


3.8 


.99900 


.99902 


.99904 


.99906 


.99908 


.99909 


.99911 


.99913 


.99915 


.99916 


3.9 


.99918 


.99920 


.99921 


.99923 


.99924 


.99926 


.99927 


.99929 


.99930 


.99932 


4.0 


.99933 


.99934 


.99936 


.99937 


.99938 


.99939 


.99941 


.99942 


.99943 


.99944 


4.1 


.99945 


.99946 


.99947 


.99948 


.99949 


.99950 


.99951 


.99952 


.99953 


.99954 


4.2 


.99955 


.99956 


.99957 


.99958 


.99958 


.99959 


.99960 


.99961 


.99962 


.99962 


4.3 


.99963 


.99964 


.99965 


.99966 


.99966 


.99967 


.99967 


.99968 


.99969 


.99969 


4.4 


.99970 


.99970 


.99971 


.99972 


.99972 


.99973 


.99973 


.99974 


.99974 


.99975 


4.5 


.99975 


.99976 


.99976 


.99977 


.99977 


.99978 


.99978 


.99979 


.99979 


.99979 


4.6 


.99980 


.99980 


.99981 


.99981 


.99981 


.99982 


.99982 


.99982 


.99983 


.99983 


4.7 


.99983 


.99984 


.99984 


.99984 


.99985 


.99985 


.99985 


.99986 


.99986 


.99986 


4.8 


.99986 


.99987 


.99987 


.99987 


.99987 


.99988 


.99988 


.99988 


.99988 


.99989 


4.9 


.99989 


.99989 


.99990 


.99990 


.99990 


.99990 


.99990 


.99990 


.99991 


.99991 


5.0 


.99991 


.99991 


.99991 


.99991 


.99992 


.99992 


.99992 


.99992 


.99992 


.99992 


5.1 


.99993 


.99993 


.99993 


.99993 


.99993 


.99993 


.99993 


.99994 


.99994 


.99994 


5.2 


.99994 


.99994 


.99994 


.99994 


.99994 


.99994 


.99995 


.99995 


.99995 


.99995 


5.3 


.99995 


.99995 


.99995 


.99995 


.99995 


.99995 


.99996 


.99996 


.99996 


.99996 


5.4 


.99996 


.99996 


.99996 


.99996 


.99996 


.99996 


.99996 


.99996 


.99997 


.99997 


5.5 


.99997 


.99997 


.99997 


.99997 


.99997 


.99997 


.99997 


.99997 


.99997 


.99997 


5.6 


.99997 


.99997 


.99997 


.99997 


.99997 


.99998 


.99998 


.99998 


.99998 


.99998 


5.7 


.99998 


.99998 


.99998 


.99998 


.99998 


.99998 


.99998 


.99998 


.99998 


.99998 


5.8 


.99998 


.99998 


.99998 


.99998 


.99998 


.99998 


.99998 


.99998 


.99998 


.99998 


5.9 


.99998 


.99999 


.99999 


.99999 


.99999 


.99999 


.99999 


.99999 


.99999 


.99999 



233 



10 

11 

12 
13 
14 

15 
16 
17 
18 
19 

20 
21 
22 
23 
24 

25 
26 
27 
28 
29 
30 

31 
32 
33 
34 

35 
36 
37 
38 
39 



nl 

1 (by definition) 

1 

2 

6 

24 

120 

720 

5040 

40,320 

362,880 

3,628,800 

39,916,800 

479,001,600 

6,227,020,800 
87,178,291,200 

1,307,674,368,000 

20,922,789,888,000 

355,687,428,096,000 

6,402,373,705,728,000 

121,645,100,408,832,000 

2,432,902,008,176,640,000 
51,090,942,171,709,440,000 

1,124,000,727,777,937,680,000 

25,852,016,738,892,566,840,000 

620,448,401,733,421,599,360,000 

15,511,210,043,336,539,984,000,000 

403,291,461,126,724,039,584,000,000 

10,888,869,450,421,549,068,768,000,000 

304,888,344,611,803,373,925,504,000,000 

8,841,761,993,742,297,843,839,616,000,000 

265,252,859,812,268,935,315,188,480,000,000 

8.22284 X 1033 
2.63131 X 1035 
8.68332 X 1036 
2.95233 X 1038 

1.03331 X 10-to 
3.71993 X 10^1 
1.37638 X 10« 
5.23023 X 10^" 
2.03979 X 1046 



n 


»! 




n 


n\ 


40 


8.15915 X 10-1^ 


80 


7.15695X10118 


41 


3.34525 X lO-") 




81 


5.79713 X 10120 


42 


1.40501 XlO^' 




82 


4.75364 X 10122 


43 


6.04153 X 10S2 




83 


3.94552 X 10124 


44 


2.65827 X lO^-i 




84 


3.31424 X 10128 


45 


1.19622 X 1056 




85 


2.81710X10128 


46 


5.50262 X 1057 




86 


2.42271 X 10130 


47 


2.58623 X 1059 




87 


2.10776 X 10132 


48 


1.24139 X 1061 




88 


1.85483 X 1013-1 


49 


6.08282 X 1062 




89 


1.65080 X 10138 


50 


3.04141 X 1061 




90 


1.48572 X 10138 


51 


1.55112 X 1066 




91 


1.35200 X 10140 


52 


8.06582 X 10^7 




92 


1.24384 X 101*2 


53 


4.27488 X 1069 




93 


1.15677 X 10144 


54 


2.30844 X lO'i 




94 


1.08737 X 10i« 


55 


1.26964 X 10^3 




95 


1.03300 X 10148 


56 


7.10999 X 1074 




96 


9.91678 X 10149 


57 


4.05269 X 10"6 




97 


9.61928 X 10151 


58 


2.35056 X 10^8 




98 


9.42689 X 10153 


59 


1.38683 X 1080 




99 


9.33262 X 10155 


60 


8.32099 X 10^1 




100 


9.33262 X 10157 


61 


5.07580 X 1083 








62 


3.14700 X 1085 








63 


1.98261 X 1087 








64 


1.26887 X 1083 








65 


8.24765 X 1090 








66 


5.44345 X 1092 








67 


3.64711X1091 








68 


2.48004 X 1096 








69 


1.71122 X 1098 








70 


1.19786 X lO'Of 








71 


8.50479 X lOioi 








72 


6.12345 X 10103 








73 


4.47012 X 10105 








74 


3.30789 X 10107 








75 


2.48091 X 10109 








76 


1.88549 X 101" 








77 


1.45183 X 10113 








78 


1.13243 X 10115 








79 


8.94618 X 10116 









234 



TABLE 

20 


GAMMA FUNCTION 

T{x) = f i^-^e-*di for 1^ a; ^2 

[For other values use the formula r(a: + 1) = x r(x)] 





X 


r{x) 


1.00 


1.00000 


1.01 


.99433 


1.02 


.98884 


1.03 


.98355 


1.04 


.97844 


1.05 


.97350 


1.06 


.96874 


1.07 


.96415 


1.08 


.95973 


1.09 


.95546 


1.10 


.95135 


1.11 


.94740 


1.12 


.94359 


1.13 


.93993 


1.14 


.93642 


1.15 


.93304 


1.16 


.92980 


1.17 


.92670 


1.18 


.92373 


1.19 


.92089 


1.20 


.91817 


1.21 


.91558 


1.22 


.91311 


1.23 


.91075 


1.24 


.90852 


1.26 


.90640 


1.26 


.90440 


1.27 


.90250 


1.28 


.90072 


1.29 


.89904 


1.30 


.89747 


1.31 


.89600 


1.82 


.89464 


1.38 


.89338 


1.34 


.89222 


t.36 


.89115 


1.86 


.89018 


1.S7 


.88931 


1.88 


.88854 


1.39 


.88785 


1.40 


.88726 


IM 


.88676 


1.42 


.88636 


1.43 


.88604 


1.44 


.88581 


1.46 


.88566 


1.46 


.88560 


1.47 


.88563 


1.48 


.88575 


1.49 


.88595 


1.50 


.88623 



X 


r(x) 


1.50 


.88623 


1.51 


.88659 


1.52 


.88704 


1.53 


.88757 


1.54 


.88818 


1.55 


.88887 


1.56 


.88964 


1.57 


.89049 


1.58 


.89142 


1.59 


.89243 


1.60 


.89352 


1.61 


.89468 


1.62 


.89592 


1.63 


.89724 


1.64 


.89864 


1.65 


.90012 


1.66 


.90167 


1.67 


.90330 


1.68 


.90500 


1.69 


.90678 


1.70 


.90864 


1.71 


.91057 


1.72 


.91258 


1.73 


.91467 


1.74 


.91683 


1.75 


.91906 


1.76 


.92137 


1.77 


.92376 


1.78 


.92623 


1.79 


.92877 


1.80 


.93138 


1.81 


.93408 


1.82 


.93685 


1.83 


.93969 


1.84 


.94261 


1.85 


.94561 


1.86 


.94869 


1.87 


.95184 


1.88 


.95507 


1.89 


.95838 


1.90 


.96177 


1.91 


.96523 


1.92 


.96877 


1.93 


.97240 


1.94 


.97610 


1.95 


.97988 


1.96 


.98374 


1.97 


.98768 


1.98 


.99171 


1.99 


.99581 


2.00 


1.00000 



235 



TABLE 



21 



BINOMIAL COEFFICIENTS 

w! _ n{n -!)■ • ■jn-k + l) _ 

klin-k)\ ~ kl 



n 
n~k 



0!=1 



Note that each number is the sum of two numbers in the row above; one of these numbers is in the same 
column and the other is in the preceding column [e.g. 56 - 35 + 21]. The arrangement is often called 
Pascal's triangle [see 3.6, page 41. 



n ^\ 


1 


2 


3 


4 


5 


6 


7 


8 


9 


1 


1 1 


















2 


1 2 


1 
















3 


1 3 


3 


1 














4 


1 4 


6 


4 


1 












5 


1 5 


10 


10 


5 


1 










6 


1 6 


15 


20 


15 


6 


1 








7 


1 7 


21 


35 


35 


21 


7 


1 






8 


1 8 


28 


56 


70 


56 


28 


8 


1 




9 


1 9 


36 


84 


126 


126 


84 


36 


9 


1 


10 


1 10 


45 


120 


210 


252 


210 


120 


45 


10 


11 


1 11 


56 


165 


330 


462 


462 


330 


166 


55 


12 


1 12 


66 


220 


495 


792 


924 


792 


495 


220 


13 


1 13 


78 


286 


715 


1287 


1716 


1716 


1287 


715 


14 


1 14 


91 


364 


1001 


2002 


3003 


3432 


3003 


2002 


15 


1 15 


105 


456 


1365 


3003 


5005 


6435 


6435 


5005 


16 


1 16 


120 


560 


1820 


4368 


8008 


11440 


12870 


11440 


17 


1 17 


136 


680 


2380 


6188 


12376 


19448 


24310 


24310 


18 


1 18 


153 


816 


3060 


8568 


18564 


31824 


43758 


48620 


19 


1 19 


171 


969 


3876 


11628 


27132 


50388 


75582 


92378 


20 


1 20 


190 


1140 


4845 


15504 


38760 


77620 


125970 


167960 


21 


1 21 


210 


1330 


5985 


20349 


54264 


116280 


203490 


293930 


22 


1 22 


231 


1540 


7315 


26334 


74613 


170544 


319770 


497420 


23 


1 23 


253 


1771 


8855 


33649 


100947 


245157 


490314 


817190 


24 


1 24 


276 


2024 


10626 


42504 


134596 


346104 


735471 


1307504 


25 


1 25 


300 


2300 


12650 


53130 


177100 


480700 


1081575 


2042975 


26 


1 26 


325 


2600 


14950 


65780 


230230 


657800 


1562275 


3124550 


27 


1 27 


351 


2925 


17550 


80730 


296010 


888030 


2220075 


4686825 


28 


1 28 


378 


3276 


20475 


98280 


376740 


1184040 


3108105 


6906900 


29 


1 29 


406 


3654 


23751 


118755 


475020 


1560780 


4292145 


10015005 


30 


1 30 


435 


4060 


27405 


142506 


593775 


2035800 


5852925 


14307150 



236 



Table 21 

(continued) 



BINOMIAL COEFFICIENTS 

n! _ n(w - 1) - ■ • (n - A: + 1) 

kl{n-k)\ " ft! 



n 
n — k 



0! = 1 



X 


10 


11 


12 


13 


14 


15 


10 


1 








- 




11 


11 


1 










12 


66 


12 


1 








13 


286 


78 


13 


1 






14 


1001 


S64 


91 


14 


1 




15 


3003 


1S6S 


455 


106 


16 


1 


16 


8008 


4368 


1820 


660 


120 


16 


17 


19448 


12376 


6188 


2380 


680 


136 


18 


43758 


31824 


18564 


8568 


3060 


816 


19 


92378 


75582 


60388 


27132 


11628 


3876 


20 


184756 


167960 


125970 


77520 


38760 


15504 


21 


352716 


352716 


293930 


203490 


116280 


54264 


22 


646646 


705432 


646646 


497420 


319770 


170544 


23 


1144066 


1362078 


1352078 


1144066 


817190 


490314 


24 


1961256 


2496144 


2704156 


2496144 


1961256 


1307504 


25 


3268760 


4457400 


5200300 


5200300 


4457400 


3268760 


26 


5311735 


7726160 


9657700 


10400600 


9657700 


7726160 


27 


8436285 


13037895 


17383860 


20058300 


20058300 


17383860 


28 


13123110 


21474180 


30421755 


37442160 


40116600 


37442160 


29 


20030010 


34597290 


51895935 


67863915 


77558760 


77558760 


30 


30045015 


54627300 


86493225 


119759850 


145422675 


155117520 



For k > 16 use the fact that 



71 

n — k 



237 



n 


«2 


n^ 


y/n 


VlOn 


y/n 


Viow 


'( 


1/n 


VlOOn 


1 


1 


1 


1.000 000 


3.162 278 


1.000 000 


2.154 435 


4.641 589 


1.000 000 


2 


4 


8 


1.414 214 


4.472 136 


1.259 921 


2.714 418 


5.848 035 


.500 000 


3 


9 


27 


1.732 051 


5.477 226 


1.442 250 


3.107 233 


6.694 330 


.333 333 


4 


16 


64 


2.000 000 


6.324 555 


1.687 401 


3.419 952 


7.368 063 


.250 000 


5 


25 


125 


2.236 068 


7.071 068 


1.709 976 


3.684 031 


7.937 005 


.200 000 


6 


36 


216 


2.449 490 


7.745 967 


1.817 121 


3.914 868 


8.434 327 


.166 667 


7 


49 


343 


2.645 751 


8.366 600 


1.912 931 


4.121 285 


8.879 040 


.142 857 


8 


64 


512 


2.828 427 


8.944 272 


2.000 000 


4.308 869 


9.283 178 


.125 000 


9 


81 


729 


3.000 000 


9.486 833 


2.080 084 


4.481 405 


9.654 894 


.111 HI 


:o 


100 


1 000 


3.162 278 


10.000 00 


2.154 435 


4.641 589 


10.000 00 


.100 000 


11 


121 


1 331 


3.316 625 


10.488 09 


2.223 980 


4.791 420 


10.322 80 


.090 909 


12 


144 


1 728 


3.464 102 


10.954 45 


2.289 428 


4.932 424 


10.626 59 


.083 333 


13 


169 


2 197 


3.005 551 


11.401 75 


2.351 335 


5.065 797 


10.913 93 


.076 923 


14 


196 


2 744 


3.741 657 


11.832 16 


2.410 142 


5.192 494 


11.186 89 


.071 429 


15 


225 


3 375 


3.872 983 


12.247 45 


2.466 212 


5.313 293 


11.447 14 


.066 667 


16 


256 


4 096 


4.000 000 


12.649 11 


2.519 842 


5.428 835 


11.696 07 


.062 500 


17 


289 


4 913 


4.123 106 


13.038 40 


2.571 282 


5.539 658 


11.934 83 


.058 824 


18 


324 


5 832 


4.242 641 


13.416 41 


2.620 741 


5.646 216 


12.164 40 


.055 556 


19 


361 


6 859 


4.358 899 


13.784 05 


2.668 402 


5.748 897 


12.385 62 


.052 632 


20 


400 


8 000 


4.472 136 


14.142 14 


2.714 418 


5.848 035 


12.599 21 


.050 000 


21 


441 


9 261 


4.582 576 


14.491 38 


2.758 924 


5.943 922 


12.805 79 


.047 619 


22 


484 


10 648 


4.690 416 


14.832 40 


2.802 039 


6.036 811 


13.005 91 


.045 455 


23 


529 


12 167 


4.795 832 


15.165 75 


2.843 867 


6.126 926 


13.200 06 


.043 478 


24 


576 


13 824 


4.898 979 


15.491 93 


2.884 499 


6.214 465 


13.388 66 


.041 667 


25 


625 


15 625 


5.000 000 


15.811 39 


2.924 018 


6.299 605 


13.572 09 


.040 000 


26 


676 


17 576 


5.099 020 


16.124 52 


2.962 496 


6.382 504 


13.750 69 


.038 462 


27 


729 


19 683 


5.196 152 


16.431 68 


3.000 000 


6.463 304 


13.924 77 


.037 037 


28 


784 


21952 


5.291 503 


16.733 20 


3.036 589 


6.542 133 


14.094 60 


.035 714 


29 


841 


24 389 


5.385 165 


17.029 39 


3.072 317 


6.619 106 


14.260 43 


.034 483 


SO 


900 


27 000 


5.477 226 


17.320 51 


3.107 233 


6.694 330 


14.422 50 


.033 333 


31 


961 


29 791 


5.567 764 


17.606 82 


3.141 381 


6.767 899 


14.581 00 


.032 258 


32 


1024 


32 768 


5.656 854 


17.888 54 


3.174 802 


6.839 904 


14.736 13 


.031 250 


33 


1 089 


35 937 


5.744 563 


18.165 90 


3.207 534 


6.910 423 


14.888 06 


.030 303 


34 


1 156 


39 304 


5.830 952 


18.439 09 


3.239 612 


6.979 532 


15.036 95 


.029 412 


35 


1225 


42 875 


5.916 080 


18.708 29 


3.271 066 


7.047 299 


15.182 94 


.028 571 


36 


1 296 


46 656 


6.000 000 


18.973 67 


3.301 927 


7.113 787 


15.326 19 


.027 778 


37 


1 369 


50 653 


6.082 763 


19.235 38 


3.332 222 


7.179 054 


15.466 80 


.027 027 


38 


1 444 


54 872 


6.164 414 


19.493 59 


3.361 975 


7.243 156 


15.604 91 


.026 316 


39 


1 521 


59 319 


6.244 998 


19.748 42 


3.391 211 


7.306 144 


15.740 61 


.025 641 


40 


1 600 


64 000 


6.324 555 


20.000 00 


3.419 952 


7.368 063 


15.874 01 


.025 000 


41 


1 681 


68 921 


6.403 124 


20.248 46 


3.448 217 


7.428 959 


16.005 21 


.024 390 


42 


1 764 


74 088 


6.480 741 


20.493 90 


3.476 027 


7.488 872 


16.134 29 


.023 810 


43 


1 849 


79 507 


6.557 439 


20.736 44 


3.503 398 


7.547 842 


16.261 33 


.023 256 


44 


1 936 


85 184 


6.633 250 


20.976 18 


3.530 348 


7.605 905 


16.386 43 


.022 727 


45 


2 025 


91 125 


6.708 204 


21.213 20 


3.556 893 


7.663 094 


16.509 64 


.022 222 


46 


2 116 


97 336 


6.782 330 


21.447 61 


3.583 048 


7.719 443 


16.631 03 


.021 739 


47 


2 209 


103 823 


6.855 655 


21.679 48 


3.608 826 


7.774 980 


16.750 69 


.021 277 


48 


2 304 


110 592 


6.928 203 


21.908 90 


3.634 241 


7.829 735 


16.868 65 


.020 833 


49 


2 401 


117 649 


7.000 000 


22.135 94 


3.659 306 


7.883 735 


16.984 99 


.020 408 


50 


2 500 


125 000 


7.071 068 


22.360 68 


3.684 031 


7.937 005 


17.099 76 


.020 000 



238 



Tabre 22 
(continued) 



SQUARES, CUBES, ROOTS AND RECIPROCALS 



n 


n2 


„3 


\n 


VlOn 


■\/n 


V^lOw 


\/lOOn 


1/n 


50 


2 500 


125 000 


7.071 068 


22.360 68 


3.684 031 


7.937 005 


17.099 76 


.020 000 


51 


2 601 


132 651 


7.141 428 


22.583 18 


3.708 430 


7.989 570 


17.213 01 


.019 608 


52 


2 704 


140 608 


7.211 103 


22.803 51 


3.732 511 


8.041 452 


17.324 78 


.019 231 


53 


2 809 


148 877 


7.280 110 


23.021 73 


3.756 286 


8.092 672 


17.435 13 


.018 868 


54 


2 916 


157 464 


7.348 469 


23.237 90 


3.779 763 


8.143 253 


17.544 11 


.018 519 


55 


3 025 


166 375 


7.416 198 


23.452 08 


3.802 952 


8.193 213 


17.651 74 


.018 182 


56 


3 136 


175 616 


7.483 315 


23.664 32 


3.825 862 


8.242 571 


17.758 08 


.017 857 


57 


3 249 


185 193 


7.549 834 


23.874 67 


3.848 501 


8.291 344 


17.863 16 


.017 544 


58 


3 364 


195 112 


7.615 773 


24.083 19 


3.870 877 


8.339 551 


17.967 02 


.017 241 


59 


3 481 


205 379 


7.681 146 


24.289 92 


3.892 996 


8.387 207 


18.069 69 


.016 949 


60 


3 600 


216 000 


7.745 967 


24.494 90 


3.914 868 


8.434 327 


18.171 21 


.016 667 


61 


3 721 


226 981 


7.810 250 


24.698 18 


3.936 497 


8.480 926 


18.271 60 


.016 393 


62 


3 844 


238 328 


7.874 008 


24.899 80 


3.957 892 


8.527 019 


18.370 91 


.016 129 


63 


3 969 


250 047 


7.937 254 


25.099 80 


3.979 057 


8.572 619 


18.469 15 


,015 873 


64 


4 096 


262 144 


8.000 000 


25.298 22 


4.000 000 


8.617 739 


18.566 36 


.015 625 


65 


4 225 


274 625 


8.062 258 


25.495 10 


4.020 726 


8.662 391 


18.662 56 


.015 385 


66 


4 356 


287 496 


8.124 038 


25.690 47 


4.041 240 


8.706 588 


18.757 77 


.015 152 


67 


4 489 


300 763 


8.185 353 


25.884 36 


4.061 548 


8.750 340 


18.852 04 


.014 925 


68 


4 624 


314 432 


8.246 211 


26.076 81 


4.081 655 


8.793 659 


18.945 36 


.014 706 


69 


4 761 


328 509 


8.306 624 


26.267 85 


4.101 566 


8.836 556 


19.037 78 


.014 493 


70 


4 900 


343 000 


8.366 600 


26.457 51 


4.121 285 


8.879 040 


19.129 31 


.014 286 


71 


5 041 


357 911 


8.426 150 


26.645 83 


4.140 818 


8.921 121 


19.219 97 


.014 085 


72 


5 184 


373 248 


8.485 281 


26.832 82 


4.160 168 


8.962 809 


19.309 79 


.013 889 


73 


5 329 


389 017 


8.544 004 


27.018 51 


4.179 339 


9.004 113 


19.398 77 


.013 699 


74 


5 476 


405 224 


8.602 325 


27.202 94 


4.198 336 


9.045 042 


19.486 95 


.013 514 


75 


5 625 


421 875 


8.660 254 


27.386 13 


4.217 163 


9.085 603 


19.574 34 


.013 333 


76 


5 776 


438 976 


8.717 798 


27.568 10 


4.235 824 


9.125 805 


19.660 95 


.013 158 


77 


5 929 


456 533 


8.774 964 


27.748 87 


4.254 321 


9.165 656 


19.746 81 


.012 987 


78 


6 084 


474 552 


8.831 761 


27.928 48 


4.272 659 


9.205 164 


19.831 92 


.012 821 


79 


6 241 


493 039 


8.888 194 


28.106 94 


4.290 840 


9.244 335 


19.916 32 


.012 6.58 


80 


6 400 


512 000 


8.944 272 


28.284 27 


4.308 869 


9.283 178 


20.000 00 


.012 500 


81 


6 561 


531 441 


9.000 000 


28.460 50 


4.326 749 


9.321 698 


20.082 99 


.012 346 


82 


6 724 


551 368 


9.055 385 


28.635 64 


4.344 481 


9.359 902 


20.165 30 


.012 195 


83 


6 889 


571 787 


9.110 434 


28.809 72 


4.362 071 


9.397 796 


20.246 94 


.012 048 


84 


7 056 


592 704 


9.165 151 


28.982 75 


4.379 519 


9.435 388 


20.327 93 


.011 905 


85 


7 225 


614 125 


9.219 544 


29.154 76 


4.396 830 


9.472 682 


20.408 28 


.011 765 


86 


7 396 


636 056 


9.273 618 


29.325 76 


4.414 005 


9.509 685 


20.488 00 


.011 628 


87 


7 569 


658 503 


9.327 379 


29.495 76 


4.431 048 


9.546 403 


20.567 10 


.011 494 


88 


7 744 


681 472 


9.380 832 


29.664 79 


4.447 960 


9.582 840 


20.645 60 


.011 364 


89 


7 921 


704 969 


9.433 981 


29.832 87 


4.464 745 


9.619 002 


20.723 51 


.011 236 


90 


8 100 


729 000 


9.486 833 


30.000 00 


4.481 405 


9.654 894 


20.800 84 


.011 111 


91 


8 281 


753 571 


9.539 392 


30.166 21 


4.497 941 


9.690 521 


20.877 59 


.010 989 


92 


8 464 


778 688 


9.591 663 


30.331 50 


4.514 357 


9.725 888 


20.953 79 


.010 870 


93 


8 649 


804 357 


9.643 651 


30.495 90 


4.530 655 


9.761 000 


21.029 44 


.010 753 


94 


8 836 


830 584 


9.695 360 


30.659 42 


4.546 836 


9.795 861 


21.104 54 


.010 638 


95 


9 025 


857 375 


9.746 794 


30.822 07 


4.562 903 


9.830 476 


21.179 12 


.010 526 


96 


9 216 


884 736 


9.797 959 


30.983 87 


4.578 857 


9.864 848 


21.253 17 


.010 417 


97 


9 409 


912 673 


9.848 858 


31.144 82 


4.594 701 


9.898 983 


21.326 71 


.010 309 


98 


9 604 


941 192 


9.899 495 


31.304 95 


4.610 436 


9.932 884 


21.399 75 


.010 204 


99 


9 801 


970 299 


9.949 874 


31.464 27 


4.626 065 


9.966 555 


21.472 29 


.010 101 


100 


10 000 


1 000 000 


10.00 000 


31.622 78 


4.641 589 


10.00 000 


21.544 35 


.010 000 



239 




COMPOUND AMOUNT: (1 + r)" 

If a principal P is deposited at interest rate r (in decimals) compounded 
annually, then at the end of n years the accumulated amount A = Pd+r)-*. 



\ 


1^. 


11% 


1|% 


2% 


2^% 


3% 


A^' 


5% 


6% 


1 


1.0100 


1.0125 


1.0150 


1.0200 


1.0250 


1.0300 


1.0400 


1.0500 


1.0600 


2 


1.0201 


1.0252 


1.0302 


1.0404 


1.0506 


1.0609 


1.0816 


1.1025 


1.1236 


3 


1.0303 


1.0380 


1.0457 


1.0612 


1.0769 


1.0927 


1.1249 


1.1576 


1.1910 


4 


1.0406 


1.0509 


1.0614 


1.0824 


1.1038 


1.1255 


1.1699 


1.2155 


1.2635 


5 


1.0510 


1.0641 


1.0773 


1.1041 


1.1314 


1.1593 


1.2167 


1.2763 


1.3382 


6 


1.0615 


1.0774 


1.0934 


1.1262 


1.1597 


1.1941 


1.2653 


1.3401 


1.4185 


7 


1.0721 


1.0909 


1.1098 


1.1487 


1.1887 


1.2299 


1.3159 


1.4071 


1.5036 


8 


1.0829 


1.1045 


1.1265 


1.1717 


1.2184 


1.2668 


1.3688 


1.4775 


1.5938 


9 


1.0937 


1.1183 


1.1434 


1.1951 


1.2489 


1.3048 


1.4233 


1.5513 


1.6895 


10 


1.1046 


1.1323 


1.1605 


1.2190 


1.2801 


1.3439 


1.4802 


1.6289 


1.7908 


11 


1.1157 


1.1464 


1.1779 


1.2434 


1.3121 


1.3842 


1.53iJ5 


1.7103 


1.8983 


12 


1.1268 


1.1608 


1.1956 


1.2682 


1.3449 


1.4258 


1.6010 


1.7959 


2.0122 


13 


1.1381 


1.1753 


1.2136 


1.2936 


1.3785 


1.4685 


1.6651 


1.8856 


2.1329 


14 


1.1495 


1.1900 


1.2318 


1.3195 


1.4130 


1.5126 


1.7317 


1.9799 


2.2609 


15 


1.1610 


1.2048 


1.2502 


1.3459 


1.4483 


1.5580 


1.8009 


2.0789 


2.3966 


16 


1.1726 


1.2199 


1.2690 


1.3728 


1.4845 


1.6047 


1.8730 


2.1829 


2.5404 


17 


1.1843 


1.2351 


1.2880 


1.4002 


1.5216 


1.6528 


1.9479 


2.2920 


2.6928 


18 


1.1961 


1.2506 


1.3073 


1.4282 


1.5597 


1.7024 


2.0258 


2.4066 


2.8543 


19 


1.2081 


1.2662 


1.3270 


1.4568 


1.5987 


1.7535 


2.1068 


2.5270 


3.0256 


20 


1.2202 


1.2820 


1.3469 


1.4859 


1.6386 


1.8061 


2.1911 


2.6533 


3.2071 


21 


1.2324 


1.2981 


1.3671 


1.5157 


1.6796 


1.8603 


2.2788 


2.7860 


3.3996 


22 


1.2447 


1.3143 


1.3876 


1.5460 


1.7216 


1.9161 


2.3699 


2.9253 


3.6035 


23 


1.2572 


1.3307 


1.4084 


1.5769 


1.7646 


1.9736 


2.4647 


3.0715 


3.8197 


24 


1.2697 


1.3474 


1.4295 


1.6084 


1.8087 


2.0328 


2.5633 


3.2251 


4.0489 


25 


1.2824 


1.3642 


1.4509 


1.6406 


1.8539 


2.0938 


2.6658 


3.3864 


4.2919 


26 


1.2953 


1.3812 


1.4727 


1.6734 


1.9003 


2.1566 


2.7725 


3.5557 


4.5494 


27 


1.3082 


1.3985 


1.4948 


1.7069 


1.9478 


2.2213 


2.8834 


3.7335 


4.8223 


28 


1.3213 


1.4160 


1.5172 


1.7410 


1.9965 


2.2879 


2.9987 


3.9201 


5.1117 


29 


1.3345 


1.4337 


1.5400 


1.7758 


2.0464 


2.3566 


3.1187 


4.1161 


5.4184 


30 


1.3478 


1.4516 


1.6631 


1.8114 


2.0976 


2.4273 


3.2434 


4.3219 


5.7435 


31 


1.3613 


1.4698 


1.5865 


1.8476 


2,1500 


2.5001 


3.3731 


4.5380 


6.0881 


32 


1.3749 


1.4881 


1.6103 


1.8845 


2,2038 


2.5751 


3.5081 


4.7649 


6.4534 


33 


1.3887 


1.5067 


1.6345 


1.9222 


2,2589 


2.6523 


3.6484 


5.0032 


6.8406 


34 


1.4026 


1.5256 


1.6590 


1.9607 


2.3153 


2.7319 


3.7943 


6.2533 


7.2510 


35 


1.4166 


1.5446 


1.6839 


1.9999 


2.3732 


2.8139 


3.9461 


5.5160 


7.6861 


36 


1.4308 


1.5639 


1.7091 


2.0399 


2.4325 


2.8983 


4.1039 


5.7918 


8.1473 


37 


1.4451 


1.5835 


1.7348 


2.0807 


2.4933 


2.9852 


4.2681 


6.0814 


8.6361 


38 


1.4595 


1.6033 


1.7608 


2.1223 


2,6557 


3.0748 


4.4388 


6.3855 


9.1543 


39 


1.4741 


1.6233 


1.7872 


2.1647 


2.6196 


3.1670 


4.6164 


6.7048 


9.7035 


40 


1.4889 


1.6436 


1.8140 


2.2080 


2.6851 


3.2620 


4.8010 


7.0400 


10.2857 


41 


1.5038 


1.6642 


1.8412 


2.2522 


2.7522 


3.3599 


4.9931 


7.3920 


10,9029 


42 


1.5188 


1.6850 


1.8688 


2.2972 


2.8210 


3.4607 


5.1928 


7.7616 


11.5570 


43 


1.5340 


1.7060 


1,8969 


2.3432 


2.8915 


3.5645 


5.4005 


8.1497 


12.2505 


44 


1.5493 


1.7274 


1.9253 


2.3901 


2.9638 


3.6715 


5.6165 


8.5572 


12.9855 


45 


1.5648 


1.7489 


1.9542 


2.4379 


3.0379 


3.7816 


5.8412 


8.9850 


13.7646 


46 


1.5805 


1.7708 


1.9835 


2.4866 


3.1139 


3.8950 


6.0748 


9.4343 


14,5905 


47 


1.5963 


1.7929 


2.0133 


2.5363 


3.1917 


4.0119 


6.3178 


9.9060 


15.4659 


48 


1.6122 


1.8154 


2.0435 


2.5871 


3.2715 


4.1323 


6.5705 


10.4013 


16.3939 


49 


1.6283 


1.8380 


2.0741 


2.6388 


3.3533 


4.2562 


6.8333 


10.9213 


17.3775 


50 


1.6446 


1.8610 


2.1052 


2.6916 


3.4371 


4.3839 


7.1067 


11.4674 


18.4202 



240 



TABLE 



24 



PRESENT VALUE OF AN AMOUNT: (1 + r) " 

The present value P which will amount to A in « years at an in- 
terest rate of r (in decimals) compounded annually is P =^ -A(l + r)". 



1% 



li% 



i^-% 



2% 



H' 



3% 



4'c 



6% 



1 
2 
3 

4 
5 

6 

7 

8 

9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 



.99010 
.98030 
.97059 
.96098 
.95147 
.94205 
.93272 
.92348 
.91434 
.90529 

.89632 
.88745 
.87866 
.86996 
.86135 
.85282 
.84438 
.83602 
.82774 
.81954 

.81143 
.80340 
.79544 
.78757 
.77977 
.77205 
.76440 
.75684 
.74934 
.74192 

.73458 
.72730 
.72010 
.71297 
.70591 
.69892 
.69200 
.68515 
.67837 
.67165 

.66500 
.65842 
.65190 
.64545 
.63905 
.63273 
.62646 
.62026 
.61412 
.60804 



.98765 
.97546 
.96342 
.95152 
.93978 
.92817 
.91672 
.90540 
.89422 
.88318 

.87228 
.86151 
.85087 
.84037 
.82999 
.81975 
.80963 
.79963 
.78976 
.78001 

.77038 
.76087 
.75147 
.74220 
.73303 
.72398 
.71505 
.70622 
.69750 
.68889 

.68038 
.67198 
.66369 
.65549 
.64740 
.63941 
.63152 
.62372 
.61602 
.60841 

.60090 
.59348 
.58616 
.57892 
.57177 
.56471 
.55774 
.55086 
.54406 
.53734 



.98522 
.97066 
.95632 
.94218 
.92826 
.91454 
.90103 
.88771 
.87459 
.86167 

.84893 
.83639 
.82403 
.81185 
.79985 
.78803 
.77639 
.76491 
.75361 
.74247 

.73150 
.72069 
.71004 
.69954 
.68921 
.67902 
.66899 
.65910 
.64936 
.63976 

.63031 
.62099 
.61182 
.60277 
.59387 
.58509 
.57644 
.56792 
.55953 
.55126 

.54312 
.53509 
.52718 
.51939 
.51171 
.50415 
.49670 
.48936 
.48213 
.47500 



.98039 
.96117 
.94232 
.92385 
.90573 
.88797 
.87056 
.85349 
.83676 
.82035 

.80426 
.78849 
.77303 
.75788 
.74301 
.72845 
.71416 
.70016 
.68643 
.67297 

.65978 
.64684 
.63416 
.62172 
.60953 
.59758 
.58586 
.57437 
.56311 
.55207 

.54125 
.53063 
.52023 
.51003 
.50003 
.49022 
.48061 
.47119 
.46195 
.45289 

.44401 
.43530 
.42677 
.41840 
.41020 
.40215 
.39427 
.38654 
.37896 
.37153 



.97561 
.95181 
.92860 
.90595 
.88385 
.86230 
.84127 
.82075 
.80073 
.78120 

.76214 
.74356 
.72542 
.70773 
.69047 
.67362 
.65720 
.64117 
.62553 
.61027 

.59539 
.58086 
.56670 
.55288 
.53939 
.52623 
.51340 
.50088 
.48866 
.47674 

.46511 
.45377 
.44270 
.43191 
.42137 
.41109 
.40107 
.39128 
.38174 
.37243 

.36335 
.35448 
.34584 
.33740 
.32917 
.32115 
.31331 
.30567 
.29822 
.29094 



.97087 
.94260 
.91514 
.88849 
.86261 
.83748 
.81309 
.78941 
.76642 
.74409 

.72242 
.70138 
.68095 
.66112 
.64186 
.62317 
.60502 
.58739 
.57029 
.55368 

.53755 
.52189 
.50669 
.49193 
.47761 
.46369 
.45019 
.43708 
.42435 
.41199 

.39999 
.38834 
.37703 
.36604 
.35538 
.34503 
.33498 
.32523 
.31575 
.30656 

.29763 
.28896 
.28054 
.27237 
.26444 
.25674 
.24926 
.24200 
.23495 
.22811 



.96154 
.92456 
.88900 
.85480 
.82193 
.79031 
.75992 
.73069 
.70259 
.67556 

.64958 
.62460 
.60057 
.57748 
.55526 
.53391 
.51337 
.49363 
.47464 
.45639 

.43883 
.42196 
.40573 
.39012 
.37512 
.36069 
.34682 
.33348 
.32065 
.30832 

.29646 
.28506 
.27409 
.26355 
.26342 
.24367 
.23430 
.22529 
.21662 
.20829 

.20028 
.19257 
.18517 
.17805 
.17120 
.16461 
.15828 
.15219 
.14634 
.14071 



.95238 
.90703 
.86384 
.82270 
.78353 
.74622 
.71068 
.67684 
.64461 
.61391 

.58468 
.55684 
.53032 
.50507 
.48102 
.45811 
.43630 
.41552 
.39573 
.37689 

.35894 
.34185 
.32557 
.31007 
.29530 
.28124 
.26785 
.25509 
.24295 
.23138 

.22036 
.20987 
.19987 
.19035 
.18129 
.17266 
.16444 
.15661 
.14915 
.14205 

.13528 
.12884 
.12270 
.11686 
.11130 
.10600 
.10095 
.09614 
.09156 
.08720 



.94340 
.89000 
.83962 
.79209 
.74726 
.70496 
.66506 
.62741 
.59190 
.55839 

.52679 
.49697 
.46884 
.44230 
.41727 
.39365 
.37136 
.35034 
.33051 
.31180 

.29416 
.27751 
.26180 
.24698 
.23300 
.21981 
.20737 
.19563 
.18456 
.17411 

.16425 
.15496 
.14619 
.13791 
.13011 
.12274 
.11579 
.10924 
.10306 
.09722 

.09172 
.08653 
.08163 
.07701 
.07265 
.06854 
.06466 
.06100 
.05755 
.05429 



241 




AMOUNT OF AN ANNUITY: 



(l + r)«-l 



If a principal P is deposited at the end of each year at interest rate r (in 
decimals) compounded annually, then at the end of n years the accumulated 

amount is P . The process is often called an annuity. 



\ 


1% 


H% 


u% 


2% 


2^% 


3% 


4fc 


5% 


6% 


1 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


2 


2.0100 


2.0125 


2.0150 


2.0200 


2-0250 


2.0300 


2.0400 


2.0500 


2.0600 


3 


3.0301 


3.0377 


3.0452 


3.0604 


3.0756 


3.0909 


3.1216 


3.1525 


3.1836 


4 


4.0604 


4.0756 


4.0909 


4.1216 


4.1525 


4.1836 


4.2465 


4.3101 


4.3746 


5 


5.1010 


5.1266 


5.1523 


5.2040 


5.2563 


5.3091 


5.4163 


5.5256 


5.6371 


6 


6.1520 


6.1907 


6.2296 


6.3081 


6.3877 


6.4684 


6.6330 


6.8019 


6.9753 


7 


7.2135 


7.2680 


7.3230 


7.4343 


7.5474 


7.6625 


7.8983 


8.1420 


8.3938 


8 


8.2857 


8.3589 


8.4328 


8.5830 


8.7361 


8.8923 


9.2142 


9.5491 


9.8975 


9 


9.3685 


9.4634 


9.5593 


9.7546 


9.9545 


10.1591 


10.5828 


11.0266 


11.4913 


10 


10.4622 


10.5817 


10.7027 


10.9497 


11.2034 


11.4639 


12.0061 


12.5779 


13.1808 


11 


11.5668 


11.7139 


11.8633 


12.1687 


12.4835 


12.8078 


13.4864 


14.2068 


14.9716 


12 


12.6825 


12.8604 


13.0412 


13.4121 


13.7956 


14.1920 


15.0258 


15.9171 


16.8699 


13 


13.8093 


14.0211 


14.2368 


14.6803 


15.1404 


15.6178 


16.62G8 


17.7130 


18.8821 


14 


14.9474 


15.1964 


15.4504 


15.9739 


16.5190 


17.0863 


18.2919 


19.5986 


21.0151 


15 


16.0969 


16.3863 


16.6821 


17.2934 


17.9319 


18.5989 


20.0236 


21.5786 


23.2760 


16 


17.2579 


17.5912 


17.9324 


18.6393 


19.3802 


20.1569 


21.8245 


23.6575 


25.6725 


17 


18.4304 


18.8111 


19.2014 


20.0121 


20.8647 


21.7616 


23.6975 


25.8404 


28.2129 


18 


19.6147 


20.0462 


20.4894 


21.4123 


22.3863 


23.4144 


25.6454 


28.1324 


30.9057 


19 


20.8109 


21.2968 


21.7967 


22.8406 


23.9460 


25.1169 


27.6712 


30.5390 


33.7600 


20 


22.0190 


22.5630 


23.1237 


24.2974 


25.5447 


26.8704 


29.7781 


33.0660 


36.7856 


21 


23.2392 


23.8450 


24.4705 


25.7833 


27.1833 


28.6765 


31.9692 


35.7193 


39.9927 


22 


24.4716 


25.1431 


25.8376 


27.2990 


28.8629 


30.5368 


34.2480 


38.5052 


43.3923 


23 


25.7163 


26.4574 


27.2251 


28.8450 


30.5844 


32.4529 


36.6179 


41.4305 


46.9958 


24 


26.9735 


27.7881 


28.6335 


30.4219 


32.3490 


34.4265 


39.0826 


44.5020 


50.8156 


25 


28.2432 


29.1354 


30.0630 


32.0303 


34.1578 


36.4593 


41.6459 


47.7271 


54.8645 


26 


29.5256 


30.4996 


31.5140 


33.6709 


36.0117 


38.5530 


44.3117 


51.1135 


59.1564 


27 


30.8209 


31.8809 


32.9867 


35.3443 


37.9120 


40.7096 


47.0842 


54.6691 


63.7058 


28 


32.1291 


33.2794 


34.4815 


37.0512 


39.8598 


42.9309 


49.9676 


58.4026 


68.5281 


29 


33.4504 


34.6954 


35.9987 


38.7922 


41.8563 


45.2189 


52.9663 


62.3227 


73.6398 


30 


34.7849 


36.1291 


37.5387 


40.5681 


43.9027 


47.5754 


66.0849 


66.4388 


79.0582 


31 


36.1327 


37.5807 


39.1018 


42.3794 


46.0003 


50.0027 


59.3283 


70.7608 


84.8017 


32 


37.4941 


39.0504 


40.6883 


44.2270 


48.1503 


52.6028 


62.7015 


75.2988 


90.8898 


33 


38.8690 


40.5386 


42.2986 


46.1116 


50.3540 


55.0778 


66.2095 


80.0638 


97.3432 


34 


40.2577 


42.0453 


43.9331 


48.0338 


52.6129 


57.7302 


69.8579 


85.0670 


104.1838 


35 


41.6603 


43.5709 


45.5921 


49.9945 


54.9282 


60.4621 


73.6522 


90.3203 


111.4348 


36 


43.0769 


45.1155 


47.2760 


51.9944 


57.3014 


63.2759 


77.5983 


95.8363 


119.1209 


37 


44.5076 


46.6794 


48.9851 


54.0343 


59.7339 


66.1742 


81.7022 


101.6281 


127.2681 


38 


45.9527 


48.2629 


50.7199 


56.1149 


62.2273 


69.1594 


85.9703 


107.7095 


135.9042 


39 


47.4123 


49.8662 


52.4807 


58.2372 


64.7830 


72.2342 


90.4091 


114.0950 


145.0585 


40 


48.8864 


51.4896 


54.2679 


60.4020 


67.4026 


75.4013 


95.0255 


120.7998 


154.7620 


41 


50.3752 


53.1332 


56.0819 


62.6100 


70.0876 


78.6633 


99.8265 


127.8398 


165.0477 


42 


51.8790 


54.7973 


57.9231 


64.8622 


72.8398 


82.0232 


104.8196 


135.2318 


175.9505 


43 


53.3978 


56.4823 


59.7920 


67.1595 


75.6608 


85.4839 


110.0124 


142.9933 


187.5076 


44 


54.9318 


58.1883 


61.6889 


69.5027 


78.5523 


89.0484 


115.4129 


151.1430 


199.7580 


4S 


56.4811 


59.9157 


63.6142 


71.8927 


81.5161 


92.7199 


121.0294 


159.7002 


212.7435 


46 


58.0459 


61.6646 


65.5684 


74.3306 


84.5540 


96.5015 


126.8706 


168.6852 


226.5081 


47 


59.6263 


63.4354 


67.5519 


76.8172 


87.6679 


100.3965 


132.9454 


178.1194 


241.0986 


48 


61.2226 


65.2284 


69.5652 


79.3535 


90.8596 


104.4084 


139.2632 


188.0254 


256.5645 


49 


62.8348 


67.0437 


71.6087 


81.9406 


94.1311 


108.5406 


145.8337 


198.4267 


272.9584 


50 


64.4632 


68.8818 


73.6828 


84.5794 


97.4843 


112.7969 


152.6671 


209.3480 


290.3359 



242 



TABLE 



26 



PRESENT VALUE OF AN ANNUITY: 



1- (l + r)-" 



An annuity in which the yearly payment at the end of each of n years is A at an 

Tl — (1 -f- r)~" 
interest rate r (in decimals) compounded annually has present value A 





1% 


U^-. 


1|% 


2% 


21% 


3% 


4% 


5% 


6% 


1 


0.9901 


0.9877 


0.9852 


0.9804 


0.9756 


0.9709 


0.9615 


0.9524 


0.9434 


2 


1.9704 


1.9631 


1.9559 


1.9416 


1.9274 


1.9135 


1.8861 


1.8594 


1.8334 


3 


2.9410 


2.9265 


2.9122 


2.8839 


2.8560 


2.8286 


2.7751 


2.7232 


2.6730 


4 


3.9020 


3.8781 


3.8544 


3.8077 


3.7620 


3.7171 


3.6299 


3.5460 


3.4651 


B 


4.8534 


4.8178 


4.7826 


4.7135 


4.6458 


4.5797 


4.4518 


4.3295 


4.2124 


6 


5.7955 


5.7460 


5.6972 


5.6014 


5.5081 


5.4172 


5.2421 


5.0757 


4.9173 


7 


6.7282 


6.6627 


6.5982 


6.4720 


6.3494 


6.2303 


6.0021 


5.7864 


5.5824 


8 


7.6517 


7.5681 


7.4859 


7.3255 


7.1701 


7.0197 


6.7327 


6.4632 


6.2098 


9 


8.5660 


8.4623 


8.3605 


8.1622 


7.9709 


7.7861 


7.4353 


7.1078 


6.8017 


10 


9.4713 


9.3455 


9.2222 


8.9826 


8.7521 


8.5302 


8.1109 


7.7217 


7.3601 


11 


10.3676 


10.2178 


10.0711 


9.7868 


9.5142 


9.2526 


8.7605 


8.3064 


7.8869 


12 


11.2551 


11.0793 


10.9075 


10.5753 


10,2578 


9.9540 


9.3851 


8.8633 


8.3838 


13 


12.1337 


11.9302 


11.7315 


11.3484 


10.9832 


10.6350 


9.9856 


9.3936 


8.8527 


14 


13.0037 


12.7706 


12.5434 


12.1062 


11.6909 


11.2961 


10.5631 


9.8986 


9.2950 


15 


13.8651 


13.6005 


13.3432 


12.8493 


12.3814 


11.9379 


11.1184 


10.3797 


9.7122 


16 


14.7179 


14.4203 


14.1313 


13.5777 


13.0550 


12.5611 


11.6523 


10.8378 


10.1059 


17 


15.5623 


15.2299 


14.9076 


14.2919 


13.7122 


13.1661 


12.1657 


11.2741 


10.4773 


18 


16.3983 


16.0295 


15.6726 


14.9920 


14.3534 


13.7535 


12.6593 


11.6896 


10.8276 


19 


17.2260 


16.8193 


16.4262 


15.6785 


14.9789 


14.3238 


13.1339 


12.0853 


11.1581 


20 


18.0456 


17.5993 


17.1686 


16.3514 


15.5892 


14.8775 


13.5903 


12.4622 


11.4699 


21 


18.8570 


18.3697 


17.9001 


17.0112 


16.1845 


15.4150 


14.0292 


12.8212 


11.7641 


22 


19.6604 


19.1306 


18.6208 


17.6580 


16.7654 


15.9369 


14.4511 


13.1630 


12.0416 


23 


20.4558 


19.8820 


19.3309 


18.2922 


17.3321 


16.4436 


14.8568 


13.4886 


12.3034 


24 


21.2434 


20.6242 


20.0304 


18.9139 


17.8850 


16.9355 


15.2470 


13.7986 


12.5504 


25 


22.0232 


21.3573 


20.7196 


19.5235 


18.4244 


17.4131 


15.6221 


14.0939 


12.7834 


26 


22.7952 


22.0813 


21.3986 


20.1210 


18.9506 


17.8768 


15.9828 


14.3752 


13.0032 


27 


23.5596 


22.7963 


22.0676 


20.7069 


19.4640 


18.3270 


16.3296 


14.6430 


13.2105 


28 


24.3164 


23.5025 


22.7267 


21.2813 


19.9649 


18.7641 


16.6631 


14.8981 


13.4062 


29 


25.0658 


24.2000 


23.3761 


21.8444 


20.4535 


19.1885 


16.9837 


15.1411 


13.5907 


30 


25.8077 


24.8889 


24.0158 


22.3965 


20.9303 


19.6004 


17.2920 


15.3725 


13.7648 


31 


26.5423 


25.5693 


24.6461 


22.9377 


21.3954 


20.0004 


17.5885 


15.5928 


13.9291 


32 


27.2696 


26.2413 


25.2671 


23.4683 


21.8492 


20.3888 


17.8736 


15.8027 


14.0840 


33 


27.9897 


26.9050 


25.8790 


23.9886 


22.2919 


20.7658 


18.1476 


16.0025 


14.2302 


34 


28.7027 


27.5605 


26.4817 


24.4986 


22.7238 


21.1318 


18.4112 


16.1929 


14.3681 


35 


29.4086 


28.2079 


27.0756 


24.9986 


23.1452 


21.4872 


18.6646 


16.3742 


14.4982 


36 


30.1075 


28.8473 


27.6607 


25.4888 


23.5563 


21.8323 


18.9083 


16.5469 


14.6210 


37 


30.7995 


29.4788 


28.2371 


25.9695 


23.9573 


22.1672 


19.1426 


16.7113 


14.7368 


38 


31.4847 


30.1025 


28.8051 


26.4406 


24.3486 


22.4925 


19.3679 


16.8679 


14.8460 


39 


32.1630 


30.7185 


29.3646 


26.9026 


24.7303 


22.8082 


19.5845 


17.0170 


14.9491 


40 


32.8347 


31.3269 


29.9158 


27.3555 


25.1028 


23.1148 


19.7928 


17.1591 


15.0463 


41 


33.4997 


31.9278 


30.4590 


27.7995 


25.4661 


23.4124 


19.9931 


17.2944 


15.1380 


42 


34.1581 


32.5213 


30.9941 


28.2348 


25.8206 


23.7014 


20.1856 


17.4232 


15.2245 


43 


34.8100 


33.1075 


31.5212 


28.6616 


26.1664 


23.9819 


20.3708 


17.5459 


15.3062 


44 


35.4555 


33.6864 


32.0406 


29.0800 


26.5038 


24.2543 


20.5488 


17.6628 


15.3832 


45 


36.0945 


34.2582 


32.5523 


29.4902 


26.8330 


24.5187 


20.7200 


17.7741 


15.4558 


46 


36.7272 


34.8229 


33.0565 


29.8923 


27.1542 


24.7754 


20.8847 


17.8801 


15.5244 


47 


37.3537 


35.3806 


33.5532 


30.2866 


27.4675 


26.0247 


21.0429 


17.9810 


15.5890 


48 


37.9740 


35.9315 


34.0426 


30.6731 


27.7732 


25.2667 


21.1951 


18.0772 


15.6500 


49 


38.5881 


36.4755 


34.5247 


31.0521 


28.0714 


25.5017 


21.3415 


18.1687 


15.7076 


50 


39.1961 


37.0129 


34.9997 


31.4236 


28.3623 


25.7298 


21.4822 


18.2559 


15.7619 



243 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


1.0000 


.9975 


.9900 


.9776 


.9604 


.9385 


.9120 


.8812 


.8463 


.8075 


1. 


.7652 


.7196 


.6711 


.6201 


.5669 


.5118 


.4554 


.3980 


.3400 


.2818 


2. 


.2239 


.1666 


.1104 


.0555 


.0025 


-.0484 


-.0968 


-.1424 


-.1850 


-.2243 


3. 


-.2601 


-.2921 


-.3202 


-.3443 


-.3643 


-.3801 


-.3918 


-.3992 


-.4026 


-.4018 


4. 


-.3971 


-.3887 


-.3766 


-.3610 


-.3423 


-.3205 


-.2961 


-.2693 


-.2404 


-.2097 


5. 


-.1776 


-.1443 


-.1103 


-.0758 


-.0412 


-.0068 


.0270 


.0599 


.0917 


.1220 


6. 


.1506 


.1773 


.2017 


.2238 


.2433 


.2601 


.2740 


.2851 


.2931 


.2981 


7. 


.3001 


.2991 


.2951 


.2882 


.2786 


.2663 


.2516 


.2346 


.2154 


.1944 


8. 


.1717 


.1475 


.1222 


.0960 


.0692 


.0419 


.0146 


-.0125 


-.0392 


-.0653 


9. 


-.0903 


-.1142 


-.1367 


-.1577 


-.1768 


-.1939 


-.2090 


-.2218 


-.2323 


-.2403 




X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


.0000 


.0499 


.0995 


.1483 


.1960 


.2423 


.2867 


.3290 


.3688 


.4059 


1. 


.4401 


.4709 


.4983 


.5220 


.5419 


.5579 


.5699 


.5778 


.5815 


.5812 


2. 


.5767 


.6683 


.5660 


.5399 


.5202 


.4971 


.4708 


.4416 


.4097 


.3754 


3. 


.3391 


.3009 


.2613 


.2207 


.1792 


.1374 


.0955 


.0538 


.0128 


-.0272 


4. 


-.0660 


-.1033 


-.1386 


-.1719 


-.2028 


-.2311 


-.2566 


-.2791 


-.2985 


-.3147 


6. 


-.3276 


-.3371 


-.3432 


-.3460 


-.3453 


-.3414 


-.3343 


-.3241 


-.3110 


-.2951 


6. 


-.2767 


-.2559 


-.2329 


-.2081 


-.1816 


-.1538 


-.1250 


-.0953 


-.0652 


-.0349 


7. 


-.0047 


.0252 


.0543 


.0826 


.1096 


.1352 


.1592 


.1813 


.2014 


.2192 


8. 


.2346 


.2476 


.2580 


.2657 


.2708 


.2731 


.2728 


.2697 


.2641 


.2559 


9. 


.2453 


.2324 


.2174 


.2004 


.1816 


.1613 


.1395 


.1166 


.0928 


.0684 



244 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


00 


-1.5342 


-1.0811 


-.8073 


-.6060 


-.4445 


-.3085 


-.1907 


-.0868 


.0056 


1. 


.0883 


.1622 


.2281 


.2865 


.3379 


.3824 


.4204 


.4520 


.4774 


.4968 


2. 


.5104 


.5183 


.5208 


.5181 


.5104 


.4981 


.4813 


.4605 


.4359 


.4079 


3. 


.3769 


.3431 


.3071 


.2691 


.2296 


.1890 


.1477 


.1061 


.0645 


.0234 


4. 


-.0169 


-.0561 


-.0938 


-.1296 


-.1633 


-.1947 


-.2235 


-.2494 


-.2723 


-.2921 


6. 


-.3085 


-.3216 


-.3313 


-.3374 


-.3402 


-.3395 


-.3354 


-.3282 


-.3177 


-.3044 


6. 


-.2882 


-.2694 


-.2483 


-.2251 


-.1999 


-.1732 


-.1452 


-.1162 


-.0864 


-.0563 


7. 


-.0259 


.0042 


.0339 


.0628 


.0907 


.1173 


.1424 


.1658 


.1872 


.2065 


8. 


.2235 


.2381 


.2501 


.2595 


.2662 


.2702 


.2715 


.2700 


.2659 


.2592 


9. 


.2499 


.2383 


.2245 


.2086 


.1907 


.1712 


.1502 


.1279 


.1045 


.0804 




X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


— eo 


-6.4590 


-3.3238 


-2.2931 


-1.7809 


-1.4715 


-1.2604 


-1.1032 


-.9781 


-.8731 


1. 


-.7812 


-.6981 


-.6211 


-.5485 


-.4791 


-.4123 


-.3476 


-.2847 


-.2237 


-.1644 


2. 


-.1070 


-.0517 


.0015 


.0523 


.1005 


.1459 


.1884 


.2276 


.2635 


.2959 


3. 


.3247 


.3496 


.3707 


.3879 


.4010 


.4102 


.4154 


.4167 


.4141 


,4078 


4. 


.3979 


.3846 


.3680 


.3484 


.3260 


.3010 


.2737 


.2445 


.2136 


.1812 


5. 


.1479 


.1137 


.0792 


.0445 


.0101 


-.0238 


-.0568 


-.0887 


-.1192 


-.1481 


6. 


-.1750 


-.1998 


-.2223 


-.2422 


-.2596 


-.2741 


-.2857 


-.2945 


-.3002 


-.3029 


7. 


-.3027 


-.2995 


-.2934 


-.2846 


-.2731 


-.2591 


-.2428 


-.2243 


-.2039 


-.1817 


8. 


-.1581 


-.1331 


-.1072 


-.0806 


-.0535 


-.0262 


.0011 


.0280 


.0544 


.0799 


9. 


.1043 


.1275 


.1491 


.1691 


.1871 


.2032 


.2171 


.2287 


.2379 


.2447 



245 




X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


1.000 


1.003 


1.010 


1.023 


1.040 


1.063 


1.092 


1.126 


1.167 


1.213 


1. 


1.266 


1.326 


1.394 


1.469 


1.553 


1.647 


1.750 


1.864 


1.990 


2.128 


2. 


2.280 


2.446 


2.629 


2.830 


3.049 


3.290 


3.553 


3.842 


4.157 


4.503 


3. 


4.881 


5.294 


5.747 


6.243 


6.785 


7.378 


8.028 


8.739 


9.517 


10.37 


4. 


11.30 


12.32 


13.44 


14.67 


16.01 


17.48 


19.09 


20.86 


22.79 


24.91 


5. 


27.24 


29.79 


32.58 


35.65 


39.01 


42.69 


46.74 


51.17 


56.04 


61.38 


6. 


67.23 


73.66 


80.72 


88.46 


96.96 


106.3 


116.5 


127.8 


140.1 


153.7 


7. 


168.6 


185.0 


202.9 


222.7 


244.3 


268.2 


294.3 


323.1 


354.7 


389.4 


8. 


427.6 


469.5 


515.6 


566.3 


621.9 


683.2 


750.5 


824.4 


905.8 


995.2 


9. 


1094 


1202 


1321 


1461 


1595 


1753 


1927 


2119 


2329 


2561 




BESSEL FUNCTIONS 

lAx) 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


.0000 


.0501 


.1005 


.1517 


.2040 


.2579 


.3137 


.3719 


.4329 


.4971 


1. 


.5652 


.6375 


.7147 


.7973 


.8861 


.9817 


1.085 


1.196 


1.317 


1.448 


2. 


1.591 


1.745 


1.914 


2.098 


2.298 


2.517 


2.755 


3.016 


3.301 


3.613 


3. 


3.953 


4.326 


4.734 


5.181 


5.670 


6.206 


6.793 


7.436 


8.140 


8.913 


4. 


9.759 


10.69 


11.71 


12.82 


14.05 


15.39 


16.86 


18.48 


20.25 


22.20 


5. 


24.34 


26.68 


29.25 


32.08 


35.18 


38.59 


42.33 


46.44 


50.95 


56.90 


6. 


61.34 


67.32 


73.89 


81.10 


89.03 


97.74 


107.3 


117.8 


129.4 


142.1 


7. 


156.0 


171.4 


188.3 


206.8 


227.2 


249.6 


274.2 


301.3 


331.1 


363.9 


8. 


399.9 


439.5 


483.0 


531.0 


583.7 


641.6 


705.4 


775.5 


852.7 


937.5 


9. 


1031 


1134 


1247 


1371 


1508 


16.58 


1824 


2006 


2207 


2428 



246 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


00 


2.4271 


1.7527 


1.3725 


1.1145 


.9244 


.7775 


.6605 


.5653 


.4867 


1. 


.4210 


.3656 


.3185 


.2782 


.2437 


.2138 


.1880 


.1655 


.1459 


.1288 


2. 


.1139 


.1008 


.08927 


.07914 


.07022 


.06235 


.05540 


.04926 


.04382 


.03901 


3. 


.03474 


.03095 


.02759 


.02461 


.02196 


.01960 


.01750 


.01563 


.01397 


.01248 


4. 


.01116 


.029980 


.028927 


.027988 


.027149 


.026400 


.025730 


.025132 


.024597 


.024119 


6. 


.023691 


.023308 


.022966 


.022659 


.022385 


.022139 


.021918 


.021721 


.021544 


.021386 


6. 


.021244 


.021117 


.021003 


.039001 


.038083 


.037259 


.036520 


.035857 


.035262 


.034728 


7. 


.034248 


.033817 


.033431 


.033084 


.032772 


.032492 


.032240 


.032014 


.031811 


.031629 


8. 


.031465 


.031317 


.031185 


.031066 


.049588 


.0*8626 


.0-*7761 


.0''6983 


.016283 


.0''5654 


9. 


.o-isoss 


.0^579 


.0^4121 


.0*3710 


.0^3339 


.o-isooe 


.0''2706 


.0''2436 


.0^2193 


.0*1975 




X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


CO 


9.8538 


4.7760 


3.0560 


2.1844 


1.6564 


1.3028 


1.0503 


.8618 


.7165 


1. 


.6019 


.5098 


.4346 


.3725 


.3208 


.2774 


.2406 


.2094 


.1826 


.1597 


2. 


.1399 


.1227 


.1079 


.09498 


.08372 


.07389 


.06528 


.05774 


.05111 


.04529 


3. 


.04016 


.03563 


.03164 


.02812 


.02500 


.02224 


.01979 


.01763 


.01571 


.01400 


4. 


.01248 


.01114 


.029938 


.028872 


.027923 


.027078 


.026325 


.025654 


.025055 


.024521 


5. 


.024045 


.023619 


.023239 


.022900 


.022597 


.022326 


.022083 


.021866 


.021673 


.021499 


6. 


.021344 


.021205 


.021081 


.039691 


.038693 


.037799 


.036998 


.036280 


.035636 


.035059 


7. 


.034542 


.034078 


.033662 


.033288 


.032953 


.032653 


.032383 


.032141 


.031924 


.031729 


8. 


.031554 


.031396 


.031255 


.031128 


.031014 


.0-'9120 


.0*8200 


.0*7374 


.0*6631 


.0*5964 


9. 


.0*5364 


.0*4825 


.0*4340 


.0*3904 


.0*3512 


.0*3160 


.0*2843 


.0*2559 


.0*2302 


.0*2072 



247 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


1.0000 


1.0000 


1.0000 


.9999 


.9995 


.9990 


.9980 


.9962 


.9936 


.9898 


1. 


.9844 


.9771 


.9676 


.9554 


.9401 


.9211 


.8979 


.8700 


.8367 


.7975 


2. 


.7517 


.6987 


.6377 


.5680 


.4890 


.4000 


.3001 


.1887 


.06511 


-.07137 


3. 


-.2214 


-.3855 


-.5644 


-.7584 


-.9680 


-1.1936 


-1.4353 


-1.6933 


-1.9674 


-2.2576 


4. 


-2.5634 


-2.8843 


-3.2195 


-3.5679 


-3.9283 


-4.2991 


-4.6784 


-5.0639 


-5.4531 


-5.8429 


5. 


-6.2301 


-6.6107 


-6.9803 


-7.3344 


-7.6674 


-7.9736 


-8.2466 


-8.4794 


-8.6644 


-8.7937 


6. 


-8.8583 


-8.8491 


-8.7561 


-8.5688 


-8.2762 


-7.8669 


-7.3287 


-6.6492 


-5.8155 


-4.8146 


7. 


-3.6329 


-2.2571 


-.6737 


1.1308 


3.1695 


5.4550 


7.9994 


10.814 


13.909 


17.293 


8. 


20.974 


24.957 


29.245 


33.840 


38.788 


43.936 


49.423 


55.187 


61.210 


67.469 


9. 


73.936 


80.576 


87.350 


94.208 


101.10 


107.95 


114.70 


121.26 


127.54 


133.43 




X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


.0000 


.022500 


.01000 


.02250 


.04000 


.06249 


.08998 


.1224 


.1599 


.2023 


1. 


.2496 


.3017 


.3587 


.4204 


.4867 


.5576 


.6327 


.7120 


.7963 


.8821 


2. 


.9723 


1.0654 


1.1610 


1.2585 


1.3575 


1.4572 


1.5669 


1.6567 


1.7529 


1.8472 


3. 


1.9376 


2.0228 


2.1016 


2.1723 


2.2334 


2.2832 


2.3199 


2.3413 


2.3454 


2.3300 


4. 


2.2927 


2.2309 


2.1422 


2.0236 


1.8726 


1.6860 


1.4610 


1.1946 


.8837 


.5251 


5. 


.1160 


-.3467 


-.8658 


-1.4443 


-2.0845 


-2.7890 


-3.5597 


-4.3986 


-5.3068 


-6.2854 


6. 


-7.3347 


-8.4545 


-9.6437 


-10.901 


-12.223 


-13.607 


-15.047 


-16.538 


-18.074 


-19.644 


7. 


-21.239 


-22.848 


-24.456 


-26.049 


-27.609 


-29.116 


-30.548 


-31.882 


-33.092 


-34.147 


8. 


-35.017 


-35.667 


-36.061 


-36.159 


-35.920 


-35.298 


-34.246 


-32.714 


-30.661 


-28.003 


9. 


-24.713 


-20.724 


-15.976 


-10.412 


-3.9693 


3.4106 


11.787 


21.218 


31.758 


43.459 



248 




BESSEL FUNCTIONS 



Ker (x) 




X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


CO 


2.4205 


1.7331 


1.3372 


1.0626 


.8559 


.6931 


.5614 


.4529 


.3625 


1. 


.2867 


.2228 


.1689 


.1235 


.08513 


.05293 


.02603 


.023691 


-.01470 


-.02966 


2. 


-.04166 


-.05111 


-.05834 


-.06367 


-.06737 


-.06969 


-.07083 


-.07097 


-.07030 


-.06894 


3. 


-.06703 


-.06468 


-.06198 


-.05903 


-.05590 


-.05264 


-.04932 


-.04597 


-.04265 


-.03937 


4. 


-.03618 


-.03308 


-.03011 


-.02726 


-.02456 


-.02200 


-.01960 


-.01734 


-.01525 


-.01330 


5. 


-.01151 


-.029865 


-.028359 


-.026989 


-.025749 


-.024632 


-.023632 


-.022740 


-.021952 


-.021258 


6. 


-.036630 


-.031295 


.033191 


.036991 


.021017 


.021278 


.021488 


.021653 


.021777 


.021866 


7. 


.021922 


.021951 


.021956 


.021940 


.021907 


.021860 


.021800 


.021731 


.021655 


.021572 


8. 


.021486 


.021397 


.021306 


.021216 


.021126 


.021037 


.039511 


.038675 


.037871 


.037102 


9. 


.036372 


.035681 


.035030 


.034422 


.033855 


.033330 


.032846 


.032402 


.031996 


.031628 




X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 


-.7864 


-.7769 


-.7581 


-.7331 


-.7038 


-.6716 


-.6374 


-.6022 


-.5664 


-.5305 


1. 


-.4960 


-.4601 


-.4262 


-.3933 


-.3617 


-.3314 


-.3026 


-.2752 


-.2494 


-.2251 


2. 


-.2024 


-.1812 


-.1614 


-.1431 


-.1262 


-.1107 


-.09644 


-.08342 


-.07167 


-.06083 


3. 


-.06112 


-.04240 


-.03458 


-.02762 


-.02145 


-.01600 


-.01123 


-.027077 


-.023487 


-.034108 


4. 


.022198 


.024386 


.026194 


.027661 


.028826 


.029721 


.01038 


.01083 


.01110 


.01121 


5. 


.01119 


.01105 


.01082 


.01051 


.01014 


.029716 


.029255 


.028766 


.028268 


.027739 


6. 


.O272I6 


.026696 


.026183 


.025681 


.025194 


.024724 


.024274 


.023846 


.023440 


.023058 


7. 


.022700 


.022366 


.022057 


.021770 


.021507 


.021267 


.021048 


.038498 


.036714 


.036117 


8. 


.033696 


.032440 


.031339 


.0*3809 


-.0M449 


-.031149 


-.031742 


-.032233 


-.032632 


-.032949 


9. 


-.033192 


-.033368 


-.033486 


-.033552 


-.033574 


-.033567 


-.033608 


-.033430 


-.033829 


-.033210 



249 



TABLE 



39 



VALUES FOR APPROXIMATE 
ZEROS OF BESSEL FUNCTIONS 



The following table lists the first few positive roots of various equations. Note that for all cases 
listed the successive large roots differ approximately hy v = 3.14159. . . . 





n = 


n = l 


71 ^ 2 


71 = 3 


71 — 4 


71 = 5 


n - 6 




2.4048 


3.8317 


5.1356 


6.3802 


7.5883 


8.7715 


9.9361 




5.5201 


7.0156 


8.4172 


9.7610 


11.0647 


12.3386 


13.5893 


J^(x) = 


8.6537 
11.7915 


10.1735 
13.3237 


11.6198 
14.7960 


13.0152 
16.2235 


14.3725 
17.6160 


15.7002 
18.9801 


17.0038 
20.3208 




14.9309 


16.4706 


17.9598 


19.4094 


20.8269 


22.2178 


23.5861 




18.0711 


19.6159 


21.1170 


22.5827 


24.0190 


25.4303 


26.8202 




0.8936 


2.1971 


3.3842 


4.5270 


5.6452 


6.7472 


7.8377 




3.9577 


5.4297 


6.7938 


8.0976 


9.3616 


10.5972 


11.8110 


Y^(x) = 


7.0861 
10.2223 


8.5960 
11.7492 


10.0235 
13.2100 


11.3965 
14.6231 


12.7301 
15.9996 


14.0338 

17.3471 


15.3136 
18.6707 




13.3611 


14.8974 


16.3790 


17.8185 


19.2244 


20.6029 


21.9583 




16.5009 


18.0434 


19.5390 


20.9973 


22.4248 


23.8265 


25.2062 




0.0000 


1.8412 


3.0542 


4.2012 


5.3176 


6.4156 


7.5013 




3.8317 


5.3314 


6.7061 


8.0152 


9.2824 


10.5199 


11.7349 


^n(«) = 


7.0156 
10.1735 


8.5363 
11.7060 


9.9695 
13.1704 


11.3459 
14.5859 


12.6819 
15.9641 


13.9872 
17.3128 


15.2682 
18.6374 




13.3237 


14.8636 


16.3475 


17.7888 


19.1960 


20.5755 


21.9317 




16.4706 


18.0155 


19.5129 


20.9725 


22.4010 


23.8036 


25.1839 




2.1971 


3.6830 


5.0026 


6.2536 


7.4649 


8.6496 


9.8148 




5.4297 


6.9415 


8.3507 


9.6988 


11.0052 


12.2809 


13.5328 


l'n(*) = 


8.5960 
11.7492 


10.1234 
13.2858 


11.5742 
14.7609 


12.9724 
16.1905 


14.3317 
17.5844 


15.6608 
18.9497 


16.9655 
20.2913 




14.8974 


16.4401 


17.9313 


19.3824 


20.8011 


22.1928 


23.5619 




18.0434 


19.5902 


21.0929 


22.5598 


23.9970 


25.4091 


26.7995 



250 



TABLE 



40 



EXPONENTIAL, SINE AND COSINE INTEGRALS 

Ei{x) = C ^du, St{x) = r ^^^du, Ci{x) =: f 



cos« 



du 



X 


Ei{x) 


Si{x) 


Ciix) 


.0 


CO 


.0000 


eo 


.5 


.5598 


.4931 


.1778 


1.0 


.2194 


.9461 


-.3374 


1.5 


.1000 


1.3247 


-.4704 


2.0 


.04890 


1.6054 


-.4230 


2.5 


.02491 


1.7785 


-.2859 


3.0 


.01305 


1.8487 


-.1196 


3.5 


.026970 


1.8331 


.0321 


4.0 


.023779 


1.7582 


.1410 


4.5 


.022073 


1.6541 


.1935 


5.0 


.021148 


1.5499 


.1900 


5.5 


.036409 


1.4687 


.1421 


6.0 


.033601 


1.4247 


.0681 


6.5 


.032034 


1.4218 


-.0111 


7.0 


.031155 


1.4546 


-.0767 


7.5 


.0''6583 


1.5107 


-.1156 


8.0 


.0''3767 


1.5742 


-.1224 


8.5 


.0''2162 


1.6296 


-.09943 


9.0 


.0*1245 


1.6650 


-.05535 


9.5 


.057185 


1.6745 


-.022678 


10.0 


.054157 


1.6583 


.04546 



251 



TABLE 



41 



LEGENDRE POIYNOMIALS Pn{x) 
[Po{x) = l, Piix) = x] 



X 


P2{^) 


P3N 


P^{^) 


Pii'^) 


.00 


-.5000 


.0000 


.3750 


.0000 


.05 


-.4963 


-.0747 


.3657 


.0927 


.10 


-.4850 


-.1475 


.3379 


.1788 


.16 


-.4663 


-.2166 


.2928 


.2523 


.20 


-.4400 


-.2800 


.2320 


.3075 


.25 


-.4063 


-.3359 


.1577 


.3397 


.80 


-.3650 


-.3825 


.0729 


.3454 


.35 


-.3163 


-.4178 


-.0187 


.3225 


.40 


-.2600 


-.4400 


-.1130 


.2706 


.46 


-.1963 


-.4472 


-.2050 


.1917 


.50 


-.1250 


-.4375 


-.2891 


.0898 


.55 


-.0463 


-.4091 


-.3690 


-.0282 


.60 


.0400 


-.3600 


-.4080 


-.1526 


.65 


.1338 


-.2884 


-.4284 


-.2705 


.70 


.2350 


-.1925 


-.4121 


-.3652 


.76 


.3438 


-.0703 


-.3501 


-.4164 


.80 


.4600 


.0800 


-.2330 


-.3995 


.85 


.6838 


.2603 


-.0506 


-.2857 


.90 


.7160 


.4725 


.2079 


-.0411 


.96 


.8538 


.7184 


.6541 


.3727 


1.00 


1.0000 


1.0000 


1.0000 


1.0000 



252 



TABLE 




LEGENDRE POLYNOMIALS P„(cosd) 
[Po(cos5) = l] 




9 


i'i{cos5) 


P2(C0Sff) 


FgCcosff) 


P4(costf) 


Psieose) 


0° 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


5° 


.9962 


.9886 


.9773 


.9623 


.9437 


10° 


.9848 


.9548 


.9106 


.8532 


.7840 


15° 


.9659 


.8995 


.8042 


.6847 


.5471 


20° 


.9397 


.8245 


.6649 


.4750 


.2715 


25° 


.9063 


.7321 


.5016 


.2465 


.0009 


30° 


.8660 


.6250 


.3248 


.0234 


-.2233 


35° 


.8192 


.5065 


.1454 


-.1714 


-.3691 


40° 


.7660 


.3802 


-.0252 


-.3190 


-.4197 


45° 


.7071 


.2500 


-.1768 


-.4063 


-.3757 


50° 


.6428 


.1198 


-.3002 


-.4275 


-.2545 


55° 


.5736 


-.0065 


-.3886 


-.3852 


-.0868 


60° 


.5000 


-.1250 


-.4375 


-.2891 


.0898 


65° 


.4226 


-.2321 


-.4452 


-.1552 


.2381 


70° 


.3420 


-.3245 


-.4130 


-.0038 


.3281 


75° 


.2588 


-.3995 


-.3449 


.1434 


.3427 


80° 


.1737 


-.4548 


-.2474 


.2659 


.2810 


85° 


.0872 


-.4886 


-.1291 


.3468 


.1577 


90° 


.0000 


-.5000 


.0000 


.3750 


.0000 



258 



TABLE 



43 



COMPLETE ELLIPTIC INTEGRALS OF FIRST AND SECOND KINDS 



K 



, _ - , E = \ yj\ ~- &2 sin^ e d9, fc = sm t& 



4- 


K 


E 


0° 


1.5708 


1.5708 


1 


1.5709 


1.5707 


2 


1.5713 


1.5703 


3 


1.5719 


1.5697 


4 


1.5727 


1.5689 


5 


1.5738 


1.5678 


6 


1.5751 


1.5665 


7 


1.5767 


1.5649 


8 


1.5785 


1.5632 


9 


1.5805 


1.5611 


10 


1.5828 


1.5589 


11 


1.5854 


1.5564 


12 


1.5882 


1.5537 


13 


1.5913 


1.5507 


14 


1.5946 


1.5476 


15 


1.5981 


1.5442 


16 


1.6020 


1.5405 


17 


1.6061 


1.5367 


18 


1.6105 


1.5326 


19 


1.6151 


1.5283 


20 


1.6200 


1.5238 


21 


1.6252 


1.5191 


22 


1.6307 


1.5141 


23 


1.6365 


1.5090 


24 


1.6426 


1.5037 


25 


1.6490 


1.4981 


26 


1.6557 


1.4924 


27 


1.6627 


1.4864 


28 


1.6701 


1.4803 


29 


1.6777 


1.4740 


30 


1.6858 


1.4675 



30° 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 



K 



E 



1.6858 
1.6941 
1.7028 
1.7119 
1.7214 
1.7312 
1.7415 
1.7522 
1.7633 
1.7748 
1.7868 
1.7992 
1.8122 
1.8256 
1.8396 
1.8541 
1.8691 
1.8848 
1.9011 
1.9180 
1.9356 
1.9539 
1.9729 
1.9927 
2.0133 
2.0347 
2.0571 
2.0804 
2.1047 
2.1300 
2.1565 



1.4675 
1.4608 
1.4539 
1.4469 
1.4397 
1.4323 
1.4248 
1.4171 
1.4092 
1.4013 
1.3931 
1.3849 
1.3765 
1.3680 
1.3594 
1.3506 
1.3418 
1.3329 
1.3238 
1.3147 
1.3055 
1.2963 
1.2870 
1.2776 
1.2681 
1.2587 
1.2492 
1.2397 
1.2301 
1.2206 
1.2111 



^ 


K 


E 


60° 


2.1565 


1.2111 


61 


2.1842 


1.2015 


62 


2.2132 


1.1920 


63 


2.2435 


1.1826 


64 


2.2754 


1.1732 


65 


2.3088 


1.1638 


66 


2.3439 


1.1545 


67 


2.3809 


1.1453 


68 


2.4198 


1.1362 


m 


2.4610 


1.1272 


70 


2.5046 


1.1184 


71 


2.5507 


1.1096 


72 


2.5998 


1.1011 


73 


2.6521 


1.0927 


74 


2.7081 


1.0844 


75 


2.7681 


1.0764 


76 


2.8327 


1.0686 


77 


2.9026 


1.0611 


78 


2.9786 


1.0538 


79 


3.0617 


1.0468 


80 


3.1534 


1.0401 


81 


3.2553 


1.0338 


82 


3.3699 


1.0278 


83 


3.5004 


1.0223 


84 


3.6519 


1.0172 


85 


3.8317 


1.0127 


86 


4.0528 


1.0086 


87 


4.3387 


1.0053 


88 


4.7427 


1.0026 


89 


5.4349 


1.0008 


90 


CO 


1.0000 



254 



TABLE 



44 



INCOMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND 

J"** dd 

y\ — k^ %m^B 





0° 


10° 


20° 


30° 


40° 


50° 


60° 


70° 


80° 


90° 


0° 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


10^ 


0.1745 


0.1746 


0.1746 


0.1748 


0.1749 


0.1751 


0.1752 


0.1753 


0.1754 


0.1754 


20° 


0.3491 


0.3493 


0.3499 


0.3508 


0.3520 


0.3533 


0.3545 


0.3565 


0.3561 


0.3564 


30<^ 


0.5236 


0.5243 


0.5263 


0.5294 


0.5334 


0.5379 


0.5422 


0.6459 


0.5484 


0.6493 


40° 


0.6981 


0.6997 


0.7043 


0.7116 


0.7213 


0.7323 


0.7436 


0.7535 


0.7604 


0.7629 


60° 


0.8727 


0.8756 


0.8842 


0.8982 


0.9173 


0.9401 


0.9647 


0.9876 


1.0044 


1.0107 


60° 


1.0472 


1.0519 


1.0660 


1.0896 


1.1226 


1.1643 


1.2126 


1.2619 


1.3014 


1.3170 


70° 


1.2217 


1.2286 


1.2495 


1.2853 


1.3372 


1.4068 


1.4944 


1.5969 


1.6918 


1.7354 


80° 


1.3963 


1.4056 


1.4344 


1.4846 


1.5597 


1.6660 


1.8125 


2.0119 


2.2653 


2.4362 


90° 


1.5708 


1.5828 


1.6200 


1.6858 


1.7868 


1.9356 


2.1565 


2.5046 


3.1534 


CO 



TABLE 



45 



INCOMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND 

J,* 
\/l - fc2 sin^ e dd, k = sin^ 




\ 


0° 


10° 


20° 


30° 


40° 


50° 


60° 


70° 


80° 


90° 


0° 


0.0000 


0.0000 


0.0000 


0.0000 


O.OOOO 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


10° 


0.1745 


0.1745 


0.1744 


0.1743 


0.1742 


0.1740 


0.1739 


0.1738 


0.1737 


0.1736 


20° 


0.3491 


0.3489 


0.3483 


0.3473 


0.3462 


0.3450 


0.3438 


0.3429 


0.3422 


0.3420 


30° 


0.5236 


0.5229 


0.5209 


0.6179 


0.5141 


0.5100 


0.5061 


0.5029 


0.5007 


0.5000 


40° 


0.6981 


0.6966 


0.6921 


0.6851 


0.6763 


0.6667 


0.6575 


0.6497 


0.6446 


0.6428 


50° 


0.8727 


0.8698 


0.8614 


0.8483 


0.8317 


0.8134 


0.7954 


0.7801 


0.7697 


0.7660 


60° 


1.0472 


1.0426 


1.0290 


1.0076 


0.9801 


0.9493 


0.9184 


0.8914 


0.8728 


0.8660 


70° 


1.2217 


1.2149 


1.1949 


1.1632 


1.1221 


1.0750 


1.0266 


0.9830 


0.9514 


0.9397 


80° 


1.3963 


1.3870 


1.3697 


1.3161 


1.2590 


1.1926 


1.1225 


1.0565 


1.0054 


0.9848 


90° 


1.5708 


1.5589 


1.5238 


1.4675 


1.3931 


1.3056 


1.2111 


1.1184 


1.0401 


1.0000 



255 



TABLE 



46 



ORDINATES OF THE 
STANDARD NORMAL CURVE 

1 



y = 



^/%^ 



-x'/2 




X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0.0 


.3989 


.3989 


.3989 


.3988 


.3986 


.3984 


.3982 


.3980 


.3977 


.3973 


0.1 


.3970 


.3965 


.3961 


.3956 


.3951 


.3945 


.3939 


.3932 


.3925 


.3918 


0.2 


.3910 


.3902 


.3894 


.3885 


.3876 


.3867 


.3857 


.3847 


.3836 


.3825 


0.3 


.3814 


.3802 


.3790 


.3778 


.3765 


.3752 


.3739 


.3725 


.3712 


.3697 


0.4 


.3683 


.3668 


.3653 


.3637 


.3621 


.3605 


.3589 


.3572 


.3555 


.3538 


0.5 


.3521 


.3503 


.3485 


.3467 


.3448 


.3429 


.3410 


.3391 


.3372 


.3352 


0.6 


.3332 


.3312 


.3292 


.3271 


.3251 


.3230 


.3209 


.3187 


.3166 


.3144 


0.7 


.3123 


.3101 


.3079 


.3056 


.3034 


.3011 


.2989 


.2966 


.2943 


.2920 


0.8 


.2897 


.2874 


.2850 


.2827 


.2803 


.2780 


.2756 


.2732 


.2709 


.2685 


0.9 


.2661 


.2637 


.2613 


.2589 


.2565 


.2541 


.2516 


.2492 


.2468 


.2444 


1.0 


.2420 


.2396 


.2371 


.2347 


.2323 


.2299 


.2275 


.2251 


.2227 


.2203 


1.1 


.2179 


.2155 


.2131 


.2107 


.2083 


.2059 


.2036 


.2012 


.1989 


.1965 


1.2 


.1942 


.1919 


.1895 


.1872 


.1849 


.1826 


.1804 


.1781 


.1758 


.1736 


1.3 


.1714 


.1691 


.1669 


.1647 


.1626 


.1604 


.1582 


.1561 


.1539 


.1518 


1.4 


.1497 


.1476 


.1456 


.1435 


.1415 


.1394 


.1374 


.1354 


.1334 


.1315 


1.5 


.1295 


.1276 


.1257 


.1238 


.1219 


.1200 


.1182 


.1163 


.1145 


.1127 


1.6 


.1109 


.1092 


.1074 


.1057 


.1040 


.1023 


.1006 


.0989 


.0973 


.0957 


1.7 


.0940 


.0925 


.0909 


.0893 


.0878 


.0863 


.0848 


.0833 


.0818 


.0804 


1.8 


.0790 


.0775 


.0761 


.0748 


.0734 


.0721 


.0707 


.0694 


.0681 


.0669 


1.9 


.0656 


.0644 


.0632 


.0620 


.0608 


.0596 


.0584 


.0573 


.0562 


.0551 


2.0 


.0540 


.0529 


.0519 


.0508 


.0498 


.0488 


.0478 


.0468 


.0459 


.0449 


2.1 


.0440 


.0431 


.0422 


.0413 


.0404 


.0396 


.0387 


.0379 


.0371 


.0363 


2.2 


.0355 


.0347 


.0339 


.0332 


.0325 


.0317 


.0310 


.0303 


.0297 


.0290 


2.3 


.0283 


.0277 


.0270 


.0264 


.0258 


.0252 


.0246 


.0241 


.0235 


.0229 


2.4 


.0224 


.0219 


.0213 


.0208 


.0203 


.0198 


.0194 


.0189 


.0184 


.0180 


2.5 


.0175 


.0171 


.0167 


.0163 


.0158 


.0154 


.0151 


.0147 


.0143 


.0139 


2.6 


.0136 


.0132 


.0129 


.0126 


.0122 


.0119 


.0116 


.0113 


.0110 


.0107 


2.7 


.0104 


.0101 


.0099 


.0096 


.0093 


.0091 


.0088 


.0086 


.0084 


.0081 


2.8 


.0079 


.0077 


.0075 


.0073 


.0071 


.0069 


.0067 


.0065 


.0063 


.0061 


2.9 


.0060 


.0058 


.0056 


.0055 


.0053 


.0051 


.0050 


.0048 


.0047 


.0046 


3.0 


.0044 


.0043 


.0042 


.0040 


.0039 


.0038 


.0037 


.0036 


.0035 


.0034 


3.1 


.0033 


.0032 


.0031 


.0030 


.0029 


.0028 


.0027 


.0026 


.0025 


.0025 


3.2 


.0024 


.0023 


.0022 


.0022 


.0021 


.0020 


.0020 


.0019 


.0018 


.0018 


3.3 


.0017 


.0017 


.0016 


.0016 


.0015 


.0015 


.0014 


.0014 


.0013 


.0013 


3.4 


.0012 


.0012 


.0012 


.0011 


.0011 


.0010 


.0010 


.0010 


.0009 


.0009 


3.5 


.0009 


.0008 


.0008 


.0008 


.0008 


.0007 


.0007 


.0007 


.0007 


.0006 


3.6 


.0006 


.0006 


.0006 


.0005 


.0005 


.0005 


.0005 


.0005 


.0005 


.0004 


3.7 


.0004 


.0004 


.0004 


.0004 


.0004 


.0004 


.0003 


.0003 


.0003 


.0003 


3.8 


.0003 


.0003 


.0003 


.0003 


.0003 


.0002 


.0002 


.0002 


.0002 


.0002 


3.9 


.0002 


.0002 


.0002 


.0002 


.0002 


.0002 


.0002 


.0002 


.0001 


.0001 



256 



TABLE 



47 



AREAS UNDER THE 
STANDARD NORMAL CURVE 

from — =o tox 

erf(a:) = -^ C e-^^^ dt 




X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0.0 


.5000 


.5040 


.5080 


.5120 


.5160 


.5199 


.6239 


.5279 


.5319 


.6359 


0.1 


.5398 


.5438 


.5478 


.5517 


.5557 


.5596 


.5636 


.5675 


.5714 


.5754 


0.2 


.5793 


.5832 


.5871 


.5910 


.5948 


.6987 


.6026 


.6064 


.6103 


.6141 


0.3 


.6179 


.6217 


.6255 


.6293 


.6331 


.6368 


.6406 


.6443 


.6480 


.6517 


0.4 


.6554 


.6591 


.6628 


.6664 


.6700 


.6736 


.6772 


.6808 


.6844 


.6879 


0.5 


.6915 


.6950 


.6985 


.7019 


.7054 


.7088 


.7123 


.7157 


.7190 


.7224 


0.6 


.7258 


.7291 


.7324 


.7357 


.7389 


.7422 


.7454 


.7486 


.7518 


.7549 


0.7 


.7580 


.7612 


.7642 


.7673 


.7704 


.7734 


.7764 


.7794 


.7823 


.7852 


0.8 


.7881 


.7910 


.7939 


.7967 


.7996 


.8023 


.8051 


.8078 


.8106 


.8133 


0.9 


.8159 


.8186 


.8212 


.8238 


.8264 


.8289 


.8315 


.8340 


.8366 


.8389 


1.0 


.8413 


.8438 


.8461 


.8485 


.8508 


.8531 


.8654 


.8577 


.8699 


.8621 


1.1 


.8643 


.8665 


.8686 


.8708 


.8729 


.8749 


.8770 


.8790 


.8810 


.8830 


1.2 


.8849 


.8869 


.8888 


.8907 


.8925 


.8944 


.8962 


.8980 


.8997 


.9015 


1.3 


.9032 


.9049 


.9066 


.9082 


.9099 


.9115 


.9131 


.9147 


.9162 


.9177 


1.4 


.9192 


.9207 


.9222 


.9236 


.9251 


.9265 


.9279 


.9292 


.9306 


.9319 


1.5 


.9332 


.9345 


.9357 


.9370 


.9382 


.9394 


.9406 


.9418 


.9429 


,9441 


1.6 


.9452 


.9463 


.9474 


.9484 


.9495 


.9506 


.9615 


.9525 


.9535 


.9545 


1.7 


.9554 


.9564 


.9573 


.9582 


.9691 


.9599 


.9608 


.9616 


.9626 


.9633 


1.8 


.9641 


.9649 


.9656 


.9664 


.9671 


.9678 


.9686 


.9693 


.9699 


.9706 


1.9 


.9713 


.9719 


.9726 


.9732 


.9738 


.9744 


.9750 


.9766 


.9761 


.9767 


2.0 


.9772 


.9778 


.9783 


.9788 


.9793 


.9798 


.9803 


.9808 


.9812 


.9817 


2.1 


.9821 


.9826 


.9830 


.9834 


.9838 


.9842 


.9846 


.9860 


.9854 


.9857 


2.2 


.9861 


.9864 


.9868 


.9871 


.9875 


.9878 


.9881 


.9884 


.9887 


.9890 


2.3 


.9893 


.9896 


.9898 


.9901 


.9904 


.9906 


.9909 


.9911 


.9913 


.9916 


2.4 


.9918 


.9920 


.9922 


.9925 


.9927 


.9929 


.9931 


.9932 


.9934 


.9936 


2.5 


.9938 


.9940 


.9941 


.9943 


.9945 


.9946 


.9948 


.9949 


.9951 


.9952 


2.6 


.9953 


.9955 


.9956 


.9957 


.9959 


.9960 


.9961 


.9962 


.9963 


.9964 


2.7 


.9965 


.9966 


.9967 


.9968 


.9969 


.9970 


.9971 


.9972 


.9973 


.9974 


2.8 


.9974 


.9975 


.9976 


.9977 


.9977 


.9978 


.9979 


.9979 


.9980 


.9981 


2.9 


.9981 


.9982 


.9982 


.9983 


.9984 


.9984 


.9985 


.9986 


.9986 


.9986 


S.O 


.9987 


.9987 


.9987 


.9988 


.9988 


.9989 


.9989 


.9989 


.9990 


.9990 


3.1 


.9990 


.9991 


.9991 


.9991 


.9992 


.9992 


.9992 


.9992 


.9993 


.9993 


3.2 


.9993 


.9993 


.9994 


.9994 


.9994 


.9994 


.9994 


.9995 


.9996 


.9995 


3.3 


.9995 


.9995 


.9995 


.9996 


.9996 


.9996 


.9996 


.9996 


.9996 


.9997 


3.4 


.9997 


.9997 


.9997 


.9997 


.9997 


.9997 


.9997 


.9997 


.9997 


.9998 


3.5 


.9998 


.9998 


.9998 


.9998 


.9998 


.9998 


.9998 


.9998 


.9998 


.9998 


3.6 


.9998 


.9998 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


3.7 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


3.8 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


3.9 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 



257 



TABLE 

48 


PERCENTILE VALUES (tp) FOR 
STUDENT'S t DISTRIBUTION 

with n degrees of freedom 

(shaded area = p) 


^ 


\ 


K 


t,> 



n 


*.995 


*.99 


*.975 


*.95 


*.90 


^.80 


t.7S 


^.70 


*.60 


tss 


1 


63.66 


31.82 


12.71 


6.31 


3.08 


1.376 


1.000 


.727 


.325 


.158 


2 


9.92 


6.96 


4.30 


2.92 


1.89 


1.061 


.816 


.617 


.289 


.142 


3 


5.84 


4.54 


3.18 


2.35 


1.64 


.978 


.765 


.584 


.277 


.137 


4 


4.60 


3.75 


2.78 


2.13 


1.53 


.941 


.741 


.569 


.271 


.134 


5 


4.03 


3.36 


2.57 


2.02 


1.48 


.920 


.727 


.559 


.267 


.132 


6 


3.71 


3.14 


2.45 


1.94 


1.44 


.906 


.718 


.553 


.265 


.131 


7 


3.50 


3.00 


2.36 


1.90 


1.42 


.896 


.711 


.549 


.263 


.130 


8 


3.36 


2.90 


2.31 


1.86 


1.40 


.889 


.706 


.546 


.262 


.130 


9 


3.25 


2.82 


2.26 


1.83 


1.38 


.883 


.703 


.543 


.261 


.129 


10 


3.17 


2.76 


2.23 


1.81 


1.37 


.879 


.700 


.542 


.260 


.129 


11 


3.11 


2.72 


2.20 


1.80 


1.36 


.876 


.697 


.540 


.260 


.129 


12 


3.06 


2.68 


2.18 


1.78 


1.36 


.873 


.695 


.539 


.259 


.128 


13 


3.01 


2.65 


2.16 


1.77 


1.35 


.870 


.694 


.538 


.259 


.128 


14 


2.98 


2.62 


2.14 


1.76 


1.34 


.868 


.692 


.537 


.258 


.128 


15 


2.95 


2.60 


2.13 


1.75 


1.34 


.866 


.691 


.536 


.258 


.128 


16 


2.92 


2.58 


2.12 


1.75 


1.34 


.865 


.690 


.535 


.258 


.128 


17 


2.90 


2.57 


2.11 


1.74 


1.33 


.863 


.689 


.534 


.257 


.128 


18 


2.88 


2.55 


2.10 


1.73 


1.33 


.862 


.688 


.534 


.257 


.127 


19 


2.86 


2.54 


2.09 


1.73 


1.33 


.861 


.688 


.533 


.257 


.127 


20 


2.84 


2.53 


2.09 


1.72 


1.32 


.860 


.687 


.533 


.257 


.127 


21 


2.83 


2.52 


2.08 


1.72 


1.32 


.859 


.686 


.532 


.257 


.127 


22 


2.82 


2.51 


2.07 


1.72 


1.32 


.858 


.686 


.532 


.256 


.127 


23 


2.81 


2.50 


2.07 


1.71 


1.32 


.858 


.685 


.532 


.256 


.127 


24 


2.80 


2.49 


2.06 


1.71 


1.32 


.857 


.685 


.531 


.256 


.127 


25 


2.79 


2.48 


2.06 


1.71 


1.32 


.856 


.684 


.531 


.256 


.127 


26 


2.78 


2.48 


2.06 


1.71 


1.32 


.856 


.684 


.531 


.256 


.127 


27 


2.77 


2.47 


2.05 


1.70 


1.31 


.855 


.684 


.531 


.256 


.127 


28 


2.76 


2.47 


2.05 


1.70 


1.31 


.855 


.683 


.530 


.256 


.127 


29 


2.76 


2.46 


2.04 


1.70 


1.31 


.854 


.683 


.530 


.256 


.127 


30 


2.75 


2.46 


2.04 


1.70 


1.31 


.854 


.683 


.530 


.256 


.127 


40 


2.70 


2.42 


2.02 


1.68 


1.30 


.851 


.681 


.529 


.255 


.126 


60 


2.66 


2.39 


2.00 


1.67 


1.30 


.848 


.679 


.527 


.254 


.126 


120 


2.62 


2.36 


1.98 


1.66 


1.29 


.845 


.677 


.526 


.254 


.126 


DO 


2.58 


2.33 


1.96 


1.645 


1.28 


.842 


.674 


.524 


.253 


.126 



Source: R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and 
Medical Research (6th edition, 1963), Table III, Oliver and Boyd Ltd., Edin- 
burgh, by permission of the authors and publishers. 



258 



TABLE 






PERCENTILE 


VALUES (v^) 


FOR 




/^-^ 


*s. 










Ar\ 




THE CH 


-SQUARE DISTRIBUTION 




/ 


^s. 


\ 








i 


Vr 






with n degrees of freedom 




/ 






^^"-^...^ 








T / 








(shaded area 


= P) 






1 


X 


2 
P 






" 


n 


^.995 


^.99 


^2 

A. 973 


^.95 


■^.90 


Y^ 
^.75 


^.50 


^.25 




^.05 


Y^ 

A. 025 


y2 


Y^ 


1 


7.88 


6.63 


5.02 


3.84 


2.71 


1.32 


.455 


102 


.0158 


.0039 


.0010 


.0002 


.0000 


2 


10.6 


9.21 


7.38 


5.99 


4.61 


2.77 


1.39 


575 


.211 


.103 


.0506 


.0201 


.0100 


3 


12.8 


11.3 


9.35 


7.81 


6.25 


4.11 


2.37 1.21 


.584 


.352 


.216 


.115 


.072 


4 


14.9 


13.3 


11.1 


9.49 


7.78 


5.39 


3.36 1.92 


1.06 


.711 


.484 


.297 


.207 


5 


16.7 


15.1 


12.8 


11.1 


9.24 


6.63 


4.35 2.67 


1.61 


1.15 


.831 


.554 


.412 


6 


18.5 


16.8 


14.4 


12.6 


10.6 


7.84 


5.35 3.45 


2.20 


1.64 


1.24 


.872 


.676 


7 


20.3 


18.5 


16.0 


14.1 


12.0 


9.04 


6.35 4.25 


2.83 


2.17 


1.69 


1.24 


.989 


8 


22.0 


20.1 


17.5 


15.5 


13.4 


10.2 


7.34 5.07 


3.49 


2.73 


2.18 


1.65 


1.34 


9 


23.6 


21.7 


19.0 


16.9 


14.7 


11.4 


8.34 5.90 


4.17 


3.33 


2.70 


2.09 


1.73 


10 


25.2 


23.2 


20.5 


18.3 


16.0 


12.5 


9.34 6.74 


4.87 


3.94 


3.25 


2.56 


2.16 


11 


26.8 


24.7 


21.9 


19.7 


17.3 


13.7 


10.3 7.58 


6.58 


4.57 


3.82 


3.05 


2.60 


12 


28.3 


26.2 


23.3 


21.0 


18.5 


14.8 


11.3 8.44 


6.30 


5.23 


4.40 


3.57 


3.07 


13 


29.8 


27.7 


24.7 


22.4 


19.8 


16.0 


12.3 9.30 


7.04 


5.89 


5.01 


4.11 


3.57 


14 


31.3 


29.1 


26.1 


23.7 


21.1 


17.1 


13.3 10.2 


7.79 


6.57 


5.63 


4.66 


4.07 


15 


32.8 


30.6 


27.5 


25.0 


22.3 


18.2 


14.3 11.0 


8.55 


7.26 


6.26 


5.23 


4.60 


16 


34.3 


32.0 


28.8 


26.3 


23.5 


19.4 


15.3 11.9 


9.31 


7.96 


6.91 


5.81 


5.14 


17 


35.7 


33.4 


30.2 


27.6 


24.8 


20.5 


16.3 12.8 


10.1 


8.67 


7.56 


6.41 


5.70 


18 


37.2 


34.8 


31.5 


28.9 


26.0 


21.6 


17.3 13.7 


10.9 


9.39 


8.23 


7.01 


6.26 


19 


38.6 


36.2 


32.9 


30.1 


27.2 


22.7 


18.3 14.6 


11.7 


10.1 


8.91 


7.63 


6.84 


20 


40.0 


37.6 


34.2 


31.4 


28.4 


23.8 


19.3 15.5 


12.4 


10.9 


9.59 


8.26 


7.43 


21 


41.4 


38.9 


35.5 


32.7 


29.6 


24.9 


20.3 16.3 


13.2 


11.6 


10.3 


8.90 


8.03 


22 


42.8 


40.3 


36.8 


33.9 


30.8 


26.0 


21.3 17.2 


14.0 


12.3 


11.0 


9.54 


8.64 


23 


44.2 


41.6 


38.1 


35.2 


32.0 


27.1 


22.3 18.1 


14.8 


13.1 


11.7 


10.2 


9.26 


24 


45.6 


43.0 


39.4 


36.4 


33.2 


28.2 


23.3 19.0 


15.7 


13.8 


12.4 


10.9 


9.89 


25 


46.9 


44.3 


40.6 


37.7 


34.4 


29.3 


24.3 19.9 


16.5 


14.6 


13.1 


11.5 


10.5 


26 


48.3 


45.6 


41.9 


38.9 


35.6 


30.4 


25.3 20.8 


17.3 


15.4 


13.8 


12.2 


11.2 


27 


49.6 


47.0 


43.2 


40.1 


36.7 


31.5 


26.3 21.7 


18.1 


16.2 


14.6 


12.9 


11.8 


28 


51.0 


48.3 


44.5 


41.3 


37.9 


32.6 


27.3 22.7 


18.9 


16.9 


15.3 


13.6 


12.5 


29 


52.3 


49.6 


45.7 


42.6 


39.1 


33.7 


28.3 23.6 


19.8 


17.7 


16.0 


14.3 


13.1 


30 


53.7 


50.9 


47.0 


43.8 


40.3 


34.8 


29.3 24.5 


20.6 


18.5 


16.8 


15.0 


13.8 


40 


66.8 


63.7 


59.3 


55.8 


51.8 


45.6 


39.3 33.7 


29.1 


26.5 


24.4 


22.2 


20.7 


50 


79.5 


76.2 


71.4 


67.5 


63.2 


56.3 


49.3 42.9 


37.7 


34.8 


32.4 


29.7 


28.0 


60 


92.0 


88.4 


83.3 


79.1 


74.4 


67.0 


59.3 52.3 


46.5 


43.2 


40.5 


37.5 


35.5 


70 


104.2 


100.4 


95.0 


90.5 


85.5 


77.6 


69.3 61.7 


55.3 


51.7 


48.8 


45.4 


43.3 


80 


116.3 


U2.3 


106.6 


101.9 


96.6 


88.1 


79.3 71.1 


64.3 


60.4 


57.2 


53.5 


51.2 


90 


128.3 


124.1 


118.1 


113.1 


107.6 


98.6 


89.3 80.6 


73.3 


69.1 


65.6 


61.8 


59.2 


100 


140.2 


135.8 


129.6 


124.3 


118.5 


109.1 


99.3 90.1 


82.4 


77.9 


74.2 


70.1 


67.3 



Source: Catherine M. Thompson, Table of percentage points of the x^ distribution, 
Biometrika, Vol. 32 (1941), by permission of the author and publisher. 



259 




9Sth PERCENTILE VALUES FOR 
THE > DISTRIBUTION 

tti = degrees of freedom for numerator 
na = degrees of freedom for denominator 
(shaded area = .95) 




\«1 

"A, 


1 


2 


3 


4 


5 


6 


8 


12 


16 


20 


30 


40 


50 


100 


OS 


1 


161.4 


199.5 


215.7 


224.6 


230.2 


234.0 


238.9 


243.9 


246.3 


248.0 


250.1 


251.1 


252.2 


253.0 


254.3 


2 


18.51 


19.00 


19.16 


19.25 


19.30 


19.33 


19.37 


19.41 


19.43 


19.45 


19.46 


19.46 


19.47 


19.49 


19.50 


3 


10.13 


9.55 


9.28 


9.12 


9.01 


8.94 


8.86 


8.74 


8.69 


8.66 


8.62 


8.60 


8.58 


8.56 


8.53 


4 


7.71 


6.94 


6.59 


6.39 


6.26 


6.16 


6.04 


6.91 


5.84 


5.80 


5.75 


5.71 


5.70 


5.66 


5.63 


5 


6.61 


5.79 


5.41 


5.19 


5.05 


4.95 


4.82 


4.68 


4.60 


4.56 


4.50 


4.46 


4.44 


4.40 


4.36 


6 


5.99 


5.14 


4.76 


4.53 


4.39 


4.28 


4.16 


4.00 


3.92 


3.87 


3.81 


3.77 


3.75 


3.71 


3.67 


7 


5.59 


4.74 


4.35 


4.12 


3.97 


3.87 


3.73 


3.57 


3.49 


3.44 


3.38 


3.34 


3.32 


3.28 


3.23 


8 


5.32 


4.46 


4.07 


3.84 


3.69 


3.58 


3.44 


3.28 


3.20 


3.15 


3.08 


3.05 


3.03 


2.98 


2.93 


9 


5.12 


4.26 


3.86 


3.63 


3.48 


3.37 


3.23 


3.07 


2.98 


2.93 


2.86 


2.82 


2.80 


2.76 


2.71 


10 


4.96 


4.10 


3.71 


3.48 


3.33 


3.22 


3.07 


2.91 


2.82 


2.77 


2.70 


2.67 


2.64 


2.59 


2.54 


11 


4.84 


3.98 


3.59 


3.36 


3.20 


3.09 


2.96 


2.79 


2.70 


2.65 


2.57 


2.53 


2.50 


2.45 


2.40 


12 


4.75 


3.89 


3.49 


3.26 


3.11 


3.00 


2.85 


2.69 


2.60 


2.54 


2.46 


2.42 


2.40 


2.35 


2.30 


13 


4.67 


3.81 


3.41 


3.18 


3.03 


2.92 


2.77 


2.60 


2.51 


2.46 


2.38 


2.34 


2.32 


2.26 


2.21 


14 


4.60 


3.74 


3.34 


3.11 


2.96 


2.85 


2.70 


2.53 


2.44 


2.39 


2.31 


2.27 


2.24 


2.19 


2.13 


15 


4.64 


3.68 


3.29 


3.06 


2.90 


2.79 


2.64 


2.48 


2.39 


2.33 


2.25 


2.21 


2.18 


2.12 


2.07 


16 


4.49 


3.63 


3.24 


3.01 


2.85 


2.74 


2.59 


2.42 


2.33 


2.28 


2.20 


2.16 


2.13 


2.07 


2.01 


17 


4.45 


3.59 


3.20 


2.96 


2.81 


2.70 


2.55 


2.38 


2.29 


2.23 


2.15 


2.11 


2.08 


2.02 


1.96 


18 


4.41 


3.55 


3.16 


2.93 


2.77 


2.66 


2.51 


2.34 


2.26 


2.19 


2.11 


2.07 


2.04 


1.98 


1.92 


19 


4.38 


3.52 


3.13 


2.90 


2.74 


2.63 


2.48 


2.31 


2.21 


2.15 


2.07 


2.02 


2.00 


1.94 


1.88 


20 


4.35 


3.49 


3.10 


2.87 


2.71 


2.60 


2.45 


2.28 


2.18 


2.12 


2.04 


1.99 


1.96 


1.90 


1.84 


22 


4.30 


3.44 


3.05 


2.82 


2.66 


2.65 


2.40 


2.23 


2.13 


2.07 


1.98 


1.93 


1.91 


1.84 


1.78 


24 


4.26 


3.40 


3.01 


2.78 


2.62 


2.51 


2.36 


2.18 


2.09 


2.03 


1.94 


1.89 


1.86 


1.80 


1.73 


26 


4.23 


3.37 


2.98 


2.74 


2.69 


2.47 


2.32 


2.15 


2.06 


1.99 


1.90 


1.85 


1.82 


1.76 


1.69 


28 


4.20 


3.34 


2.95 


2.71 


2.56 


2.45 


2.29 


2.12 


2.02 


1.96 


1.87 


1.81 


1.78 


1.72 


1.65 


30 


4.17 


3.32 


2.92 


2.69 


2.53 


2.42 


2.27 


2.09 


1.99 


1.93 


1.84 


1.79 


1.76 


1.69 


1.62 


40 


4.08 


3.23 


2.84 


2.61 


2.45 


2.34 


2.18 


2.00 


1.90 


1.84 


1.74 


1.69 


1.66 


1.59 


l.Bl 


50 


4.03 


3.18 


2.79 


2.56 


2.40 


2.29 


2.13 


1.96 


1.86 


1.78 


1.69 


1.63 


1.60 


1.62 


1.44 


60 


4.00 


3.15 


2.76 


2.53 


2.37 


2.25 


2.10 


1.92 


1.81 


1.75 


1.65 


1.59 


1.56 


1.48 


1.39 


70 


3.98 


3.13 


2.74 


2.50 


2.35 


2.23 


2.07 


1.89 


1.79 


1.72 


1.62 


1.56 


1.63 


1.46 


1.35 


80 


3.96 


3.11 


2.72 


2.48 


2.33 


2.21 


2.05 


1.88 


1.77 


1.70 


1.60 


1.54 


1.51 


1.42 


1.32 


100 


3.94 


3.09 


2.70 


2.46 


2.30 


2.19 


2.03 


1.85 


1.75 


1.68 


1.57 


1.51 


1.48 


1.39 


1.28 


150 


3.91 


3.06 


2.67 


2.43 


2.27 


2.16 


2.00 


1.82 


1.71 


1.64 


1.54 


1.47 


1.44 


1.34 


1.22 


200 


3.89 


3.04 


2.65 


2.41 


2.26 


2.14 


1.98 


1.80 


1.69 


1.62 


1.52 


1.45 


1.42 


1.32 


1.19 


400 


3.86 


3.02 


2.62 


2.39 


2.23 


2.12 


1.96 


1.78 


1.67 


1.60 


1.49 


1.42 


1.38 


1.28 


1.13 


CO 


3.84 


2.99 


2.60 


2.37 


2.21 


2.09 


1.94 


1.75 


1.64 


1.57 


1.46 


1.40 


1.32 


1.24 


1.00 



Source: G. W. Snedecor and W. G. Cochran, Statistical Methods (6th edition, 1967), Iowa 
State University Press, Ames, Iowa, by permission of the authors and publisher. 



260 





99th PERCENTILE VALUES FOR 






TABLE 

51 


THE F DISTRIBUTION 

ni = degrees of freedom for numerator 
«2 = degrees of freedom for denominator 




/"^^ 






^89 



W2\ 


1 


2 


3 


4 


5 


6 


8 


12 


16 


20 


30 


40 


50 


100 


CO 


1 


4052 


4999 


5403 


5625 


5764 


5859 


5981 


6106 


6169 


6208 


6258 


6286 


6302 


6334 


6366 


2 


98.49 


99.01 


99.17 


99.25 


99.30 


99.33 


99.36 


99.42 


99.44 


99.45 


99.47 


99.48 


99.48 


99.49 


99.50 


3 


34.12 


30.81 


29.46 


28.71 


28.24 


27.41 


27.49 


27.05 


28.63 


26.69 


26.50 


26.41 


26.35 


26.23 


26.12 


4 


21.20 


18.00 


16.69 


15.98 


15.52 


15.21 


14.80 


14.37 


14.15 


14.02 


13.83 


13.74 


13.69 


13.57 


13.46 


5 


16.26 


13.27 


12.06 


11.39 


10.97 


10.67 


10.27 


9.89 


9.68 


9.65 


9.38 


9.29 


9.24 


9.13 


9.02 


6 


13.74 


10.92 


9.78 


9.15 


8.75 


8.47 


8.10 


7.72 


7.52 


7.39 


7.23 


7.14 


7.09 


6.99 


6.88 


7 


12.25 


9.55 


8.45 


7.85 


7.46 


7.19 


6.84 


6.47 


6.27 


6.15 


5.98 


5.90 


5.85 


5.75 


5.66 


8 


11.26 


8.65 


7.59 


7.01 


6.63 


6.37 


6.03 


5.67 


5.48 


5.36 


5.20 


5.11 


5.06 


4.96 


4.86 


9 


10.56 


8.02 


6.99 


6.42 


6.06 


5.80 


5.47 


5.11 


4.92 


4.80 


4.64 


4.56 


4.51 


4.41 


4.31 


10 


10.04 


7.56 


6.55 


5.99 


5.64 


5.39 


5.06 


4.71 


4.52 


4.41 


4.25 


4.17 


4.12 


4.01 


3.91 


11 


9.05 


7.20 


6.22 


5.67 


5.32 


5.07 


4.74 


4.40 


4.21 


4.10 


3.94 


3.86 


3.80 


3.70 


3.60 


12 


9.33 


6.93 


5.95 


5.41 


5.06 


4.82 


4.50 


4.16 


3.98 


3.86 


3.70 


3.61 


3.56 


3.46 


3.36 


13 


9.07 


6.70 


5.74 


5.20 


4.86 


4.62 


4.30 


3.96 


3.78 


3.67 


3.51 


3.42 


3.37 


3.27 


3.16 


14 


8.86 


6.51 


5.56 


5.03 


4.69 


4.46 


4.14 


3.80 


3.62 


3.51 


3.34 


3.26 


3.21 


3.11 


3.00 


15 


8.68 


6.36 


5.42 


4.89 


4.66 


4.32 


4.00 


3.67 


3.48 


3.36 


3.20 


3.12 


3.07 


2.97 


2.87 


16 


8.53 


6.23 


5.29 


4.77 


4.44 


4.20 


3.89 


3.55 


3.37 


3.25 


3.10 


3.01 


2.96 


2.86 


2.75 


17 


8.40 


6.11 


5.18 


4.67 


4.34 


4.10 


3.79 


3.45 


3.27 


3.16 


3.00 


2.92 


2.86 


2.76 


2.65 


18 


8.28 


6.01 


5.09 


4.58 


4.25 


4.01 


3.71 


3.37 


3.19 


3.07 


2.91 


2.83 


2.78 


2.68 


2.57 


19 


8.18 


6.93 


5.01 


4.50 


4.17 


3.94 


3.63 


3.30 


3.12 


3.00 


2.84 


2.76 


2.70 


2.60 


2.49 


20 


8.10 


6.86 


4.94 


4.43 


4.10 


3.87 


3.56 


3.23 


3.05 


2.94 


2.77 


2.69 


2.63 


2.53 


2.42 


22 


7.94 


5.72 


4.82 


4.31 


3.99 


3.76 


3.45 


3.12 


2.94 


2.83 


2.67 


2.58 


2.53 


2.42 


2.31 


24 


7.82 


5.61 


4.72 


4.22 


3.90 


3.67 


3.36 


3.03 


2.85 


2.74 


2.58 


2.49 


2.44 


2.33 


2.21 


26 


7.72 


5.53 


4.64 


4.14 


3.82 


3.59 


3.29 


2.96 


2.77 


2.66 


2.50 


2.41 


2.36 


2.25 


2.13 


28 


7.64 


5.45 


4.57 


4.07 


3.76 


3.53 


3.23 


2.90 


2.71 


2.60 


2.44 


2.35 


2.30 


2.18 


2.06 


30 


7.56 


5.39 


4.51 


4.02 


3.70 


3.47 


3.17 


2.84 


2.66 


2.55 


2.38 


2.29 


2.24 


2.13 


2.01 


40 


7.31 


5.18 


4.31 


3.83 


3.51 


3.29 


2.99 


2.66 


2.49 


2.37 


2.20 


2.11 


2.05 


1.94 


1.81 


50 


7.17 


5.06 


4.20 


3.72 


3.41 


3.18 


2.88 


2.56 


2.39 


2.26 


2.10 


2.00 


1.94 


1.82 


1.68 


60 


7.08 


4.98 


4.13 


3.65 


3.34 


3.12 


2.82 


2.50 


2.32 


2.20 


2.03 


1.93 


1.87 


1.74 


1.60 


70 


7.01 


4.92 


4.08 


3.60 


3.29 


3.07 


2.77 


2.45 


2.28 


2.15 


1.98 


1.88 


1.82 


1.69 


1.53 


80 


6.96 


4.88 


4.04 


3.56 


3.25 


3.04 


2.74 


2.41 


2.24 


2.11 


1.94 


1.84 


1.78 


1.65 


1.49 


100 


6.90 


4.82 


3.98 


3.51 


3.20 


2.99 


2.69 


2.36 


2.19 


2.06 


1.89 


1.79 


1.73 


1.59 


1.43 


150 


6.81 


4.76 


3.91 


3.44 


3.14 


2.92 


2.62 


2.30 


2.12 


2.00 


1.83 


1.72 


1.66 


1.51 


1.33 


200 


6.76 


4.71 


3.88 


3.41 


3.11 


2.90 


2.60 


2.28 


2.09 


1.97 


1.79 


1.69 


1.62 


1.48 


1.28 


400 6.70 


4.66 


3.83 


3.36 


3.06 


2.85 


2.55 


2.23 


2.04 


1.92 


1.74 


1.64 


1.57 


1.42 


1.19 


6.64 


4.60 


3.78 


3.32 


3.02 


2.80 


2.51 


2.18 


1.99 


1.87 


1.69 


1.59 


1.52 


1.36 


1.00 



Source: G. W. Snedecor and W. G. Cochran, Statistical Methods {6th edition, 1967), Iowa 
State University Press, Ames, Iowa, by permission of the authors and publisher. 



261 



9 




51772 


74640 


42331 


29044 


46621 


62898 


93582 


04186 


19640 


87056 


24033 


23491 


83587 


06568 


21960 


21387 


76105 


10863 


97453 


90581 


45939 


60173 


52078 


25424 


11645 


55870 


56974 


37428 


93507 


94271 


30586 


02133 


75797 


45406 


31041 


86707 


12973 


17169 


88116 


42187 


03585 


79353 


81938 


82322 


96799 


85659 


36081 


50884 


14070 


74950 


64937 


03355 


95863 


20790 


65304 


55189 


00745 


65253 


11822 


15804 


15630 


64759 


51135 


98527 


62586 


41889 


25439 


88036 


24034 


67283 


09448 


56301 


57683 


30277 


94623 


85418 


68829 


06652 


41982 


49159 


21631 


91157 


77331 


60710 


52290 


16835 


48653 


71590 


16159 


14676 


91097 


17480 


29414 


06829 


87843 


28195 


27279 


47152 


35683 


47280 


50532 


25496 


95652 


42457 


73547 


76552 


50020 


24819 


52984 


76168 


07136 


40876 


79971 


54195 


25708 


51817 


36732 


72484 


94923 


75936 


27989 


64728 


10744 


08396 


56242 


90985 


28868 


99431 


60995 


20507 


85184 


73949 


36601 


46253 


00477 


25234 


09908 


36574 


72139 


70186 


54398 


21154 


97810 


36764 


32869 


11785 


55261 


59009 


38714 


38723 


65544 


34371 


09591 


07839 


58892 


92843 


72828 


91341 


84821 


63886 


08263 


65952 


85762 


64236 


39238 


18776 


84303 


99247 


46149 


03229 


39817 


67906 


48236 


16057 


81812 


15815 


63700 


85915 


19219 


45943 


62257 


04077 


79443 


95203 


02479 


30763 


92486 


54083 


23631 


05825 


53298 


90276 


62545 


21944 


16530 


03878 


07516 


95715 


02526 


33537 



262 



Index of Special Symbols and Notations 

The following list shows special symbols and notations used in this book together with pages on which 
they are defined or first appear. Cases where a symbol has more than one meaning wiU be clear from 



the context. 



Symbols 

Ber„(a:), Bei„(a!) 140 

B{m,n) beta function, 103 

B„ Bernoulli numbers, 114 

C{x) Fresnel cosine integral, 184 

Ci{x) cosine integral, 184 

e natural base of logarithms, 1 

61,62,63 unit vectors in curvilinear coordinates, 124 

erf (x) error function, 183 

erfc (x) complementary error function, 183 

E = E{k, it/2) complete elliptic integral of second kind, 179 

Eik, ,p) incomplete elliptic integral of second kind, 179 

EHx) exponential integral, 183 

E„ Euler numbers, 114 

F(a, b; c; x) hypergeometric function, 160 

F{k, 0) incomplete elliptic integral of first kind, 179 

y, ^-1 Fourier transform and inverse Fourier transform, 175, 176 

^1, fh, Aa scale factors in curvilinear coordinates, 124 

Hnix) Hermite polynomials, 151 

H^l\x), H'-1\x) Hankel functions of first and second kind, 138 

i imaginary unit, 21 

i, j,k unit vectors in rectangular coordinates, 117 

/„(x) modified Bessel function of first kind, 138 

Jf^{x) Bessel function of first kind, 136 

K = F{k.Trl2) complete elliptic integral of first kind, 179 

Ker„ (a:), Kei„ {x) 140 

K^{x) modified Bessel function of second kind, 139 

In X or loge X natural logarithm of x, 24 

log X OT logio X common logarithm of x, 23 

L^{x) Laguerre polynomials, 153 

Ln {«) associated Laguerre polynomials, 155 

^, ^-1 Laplace transform and inverse Laplace transform, 161 

P^{x) Legendre polynomials, 146 

^^(2:) associated Legendre functions of first kind, 149 

Qnix) Legendre functions of second kind, 148 

Q"(x) associated Legendre functions of second kind, 150 

r cylindrical coordinate, 49 

r polar coordinate, 22, 36 

r spherical coordinate, 50 

Six) Fresnel sine integral, 184 

Si(x) sine integral, 183 

r„(a:) Chebyshev polynomials of first kind, 157 
U„{x) Chebyshev polynomials of second kind, 158 
YAx) Bessel function of second kind, 136 



263 



264 



INDEX OF SPECIAL SYMBOLS AND NOTATIONS 



Greek Symbols 



y Euler's constant, 1 

r{x) gamma function, 1, 101 

f(a;) Riemann zeta function, 184 

6 cylindrical coordinate, 49 

e polar coordinate, 22, 36 



e spherical coordinate, 50 

«■ 1 

•p spherical coordinate, 60 

*(p) the sum 1 + - + ^ + ■ ■ 

'^{x) probability distribution function, 189 



+ -, *(0) = 0, 137 



Notations 

A = B A equals fi or A is equal to B 

A > B A is greater than B [or B is less than A] 

A < B A is less than B [or B is greater than A] 

A ^ £ A is greater than or equal to B 

A '^ B A is less than or equal to B 

A "^ B A is approximately equal to B 

A -^ B A is asymptotic to B or A/B approaches 1, 102 

A if A ^ 



\A\ absolute value of A = 
n \ factorial n, 3 



'' = g=/». 



y = 



_ d^y_ 



dx2 



= f"{x), etc. 



Dp = 



dxP 
dy 



df sf d-^f 



, etc. 



dx' dy' dx dy 

d{x. y, z) 



a(ui,M2,«3) 



/ 



X 



f{x) dx 
\'dt 



A>B 

AXB 

V 

V2 = V • V 

V* = V2{V2) 



-A if A ^ 



binomial coefficients, 3 

derivatives of y or f{x) witti respect to x, 53, 55 

pth derivative with respect to x, 55 
differential of y, 55 
partial derivatives, 56 

Jacobian, 125 
indefinite integral, 67 

definite integral, 94 

line integral of A along C, 121 

dot product of A and B, 117 
cross product of A and B, 118 

del operator, 119 
Laplacian operator, 120 
biharmonic operator, 120 



INDEX 



Addition formulas, for Bessel functions, 145 

for elliptic functions, 180 

for Hermite polynomials, 152 

for hyperbolic functions, 27 

for trigonometric functions, 15 
Agnesi, witch of, 43 

Algebraic equations, solutions of, 32, 33 
Amplitude, of complex number, 22 

of elliptic integral, 179 
Analytic geometry, plane [see Plane analytic 

geometry]; solid [see Solid analytic geometry] 
Angle between lines, in a plane, 35 

in space, 47 
Annuity, amount of, 201, 242 

present value of, 243 
Anti-derivative, 57 
Antilogarithms, common, 23, 195, 204, 205 

natural or Napierian, 24, 226, 227 
Archimedes, spiral of, 45 
Area integrals, 122 
Argand diagram, 22 
Arithmetic-geometric series, 107 
Arithmetic mean, 185 
Arithmetic series, 107 
Associated Laguerre polynomials, 155, 156 
[see also Laguerre polynomials] 

generating function for, 155 

orthogonal series for, 156 

orthogonality of, 156 

recurrence formulas for, 156 

special, 155 

special results involving, 156 
Associated Legendre functions, 149, 150 [see also 
Legendre functions] 

generating function for, 149 

of the first kind, X49 

of the second kind, 150 

orthogonal series for, 150 

orthogonality of, 150 

recurrence formulas for, 149 

special, 149 
Associative law, 117 
Asymptotes of hyperbola, 39 
Asymptotic expansions or formulas, for Bernoulli 
numbers, 115 

for Bessel functions, 143 

for gamma function, 102 

Base of logarithms, 23 

change of, 24 
Ber and Bei functions, 140, 141 

definition of, 140 

differential equation for, 141 

graphs of, 141 
Bernoulli numbers, 98, 107, 114, 116 

asymptotic formula for, 115 

definition of, 114 

relationship to Euler numbers, 115 

series involving, 115 

table of first few, 114 



Bernoulli's differential equation, 104 
Bessel functions, 136-145 

addition formulas for, 145 

asymptotic expansions of, 143 

definite integrals involving, 142, 143 

generating functions for, 137, 139 

graphs of, 141 

indefinite integrals involving, 142 

infinite products for, 188 

integral representations for, 143 

modified [see Modified Bessel functions] 

of first kind of order n, 136, 137 

of order half an odd integer, 138 

of second kind of order n, 136, 137 

orthogonal series for, 144, 145 

recurrence formulas for, 137 

tables of, 244-249 

zeros of, 250 
Bessel's differential equation, 106, 136 

general solution of, 106, 137 

transformed, 106 
Bessel's modified differential equation, 138 

general solution of, 139 
Beta function, 103 

relationship of to gamma function, 103 
Biharmonic operator, 120 

in curvilinear coordinates, 125 
Binomial coefficients, 3 

properties of, 4 

table of values for, 236, 237 
Binomial distribution, 189 
Binomial formula, 2 
Binomial series, 2, 110 
Bipolar coordinates, 128, 129 

Laplacian in, 128 
Branch, principal, 17 
Briggsian logarithms, 23 

Cardioid, 41, 42, 44 

Cassini, ovals of, 44 

Catalan's constant, 181 

Catenary, 41 

Cauchy or Euler differential equation, 105 

Cauchy-Schwarz inequality, 185 

for integrals, 186 
Cauchy's form of remainder in Taylor series, 110 
Chain rule for, derivatives, 53 
Characteristic, 194 
Chebyshev polynomials, 157-159 

generating functions for, 157, 158 

of first kind, 157 

of second kind, 158 

orthogonality of, 158, 159 

orthogonal series for, 158, 159 

recursion formulas for, 158, 159 

relationships involving, 159 

special, 157, 158 

special values of, 157, 159 
Chebyshev's differential equation, 157 

general solution of, 169 



265 



266 



INDEX 



Chebyshev's inequality, 186 
Chi square distribution, 189 
percentile values for, 259 
Circle, area of, 6 
equation of, 37 
involute of, 43 
perimeter of, 6 
sector of [sec Sector of circle] 
segment of [see Segment of circle] 
Cissoid of Diodes, 45 

Common antilogarithms, 23, 195, 204, 205 
sample problems involving, 195 
table of, 204, 205 
Common logarithms, 23, 194, 202, 203 
computations using, 196 
sample problems involving, 194 
table of, 202, 203 
Commutative law, for dot products, 118 

for vector addition, 117 
Complement, 20 

Complementary error function, 183 
Complex conjugate, 21 
Complex inversion formula, 161 
Complex numbers, 21, 22, 25 
addition of, 21 
amplitude of, 22 
conjugate, 21 
definitions involving, 21 
division of, 21, 25 
graphs of, 22 
imaginary part of, 21 
logarithms of, 25 
modulus of, 22 
multiplication of, 21, 26 
polar form of, 22, 25 
real part of, 21 
roots of, 22, 25 
subtraction of, 21 
vector representation of, 22 
Components of a vector, 117 
Component vectors, 117 
Compound amount, table of, 240 
Cone, elliptic, 51 

right circular [see Right circular cone] 
Confocal ellipses, 127 
ellipsoidal coordinates, 130 
hyperbolas, 127 
parabolas, 126 

paraboloidal coordinates, 130 
Conical coordinates, 129 

Laplacian in, 129 
Conies, 37 [see also Ellipse, Parabola, Hyperbola] 
Conjugate, complex, 21 
Constant of integration, 57 
Convergence, interval of, 110 

of Fourier series, 131 
Convergence factors, table of, 192 
Coordinate curves, 124 

system, 11 
Coordinates, curvilinear, 124-130 
cylindrical, 49, 126 
polar, 22, 36 
rectangular, 36, 117 



Coordinates, curvilinear (cont.) 
rotation of, 36, 49 
special orthogonal, 126-130 
spherical, 50, 126 
transformation of, 36, 48, 49 

translation of, 36, 49 
Cosine integral, 184 
Fresnel, 184 

table of values for, 251 
Cosines, law of for plane triangles, 19 

law of for spherical triangles, 19 
Counterclockwise, 11 
Cross or vector product, 118 
Cube, duplication of, 45 
Cube roots, table of, 238, 239 
Cubes, table of, 238, 239 
Cubic equation, solution of, 32 
Curl, 120 

in curvilinear coordinates, 125 
Curtate cycloid, 42 
Curves, coordinate, 124 

special plane, 40-45 
Curvilinear coordinates, 124, 125 

orthogonal, 124-130 
Cycloid, 40, 42 

curtate, 42 

prolate, 42 
Cylinder, elliptic, 51 

lateral surface area of, 8, 9 

volume of, 8, 9 
Cylindrical coordinates, 49, 126 

Laplacian in, 126 

Definite integrals, 94-100 

approximate formulas for, 95 

definition of, 94 

general formulas involving, 94, 95 

table of, 95-100 
Degrees, 1, 199, 200 

conversion of to radians, 199, 200, 223 

relationship of to radians, 12, 199, 200 
Del operator, 119 

miscellaneous formulas involving, 120 
Delta function, 170 
DeMoivre's theorem, 22, 25 
Derivatives, 53-56 [see also Differentiation] 

anti-, 57 

chain rule for, 53 

definition of, 53 

higher, 55 

of elliptic functions, 181 

of exponential and logarithmic functions, 54 

of hyperbolic and inverse hyperbolic 
functions, 54, 55 

of trigonometric and inverse trigonometric 
functions, 64 

of vectors, 119 

partial, 56 
Descartes, folium of, 43 

Differential equations, solutions of basic, 104-106 
Differentials, 55 

rules for, 56 
Differentiation, 53 [see also Derivatives] 



INDEX 



267 



Differentiation (cont.) 

general rules for, 63 

of integrals, 95 
Diodes, cissoid of, 45 
Direction cosines, 46, 47 

numbers, 46, 48 
Directrix, 37 
Discriminant, 32 
Distance, between two points in a plane, 34 

between two points in space, 46 

from a point to a line, 35 

from a point to a plane, 48 
Distributions, probability, 189 
Distributive law, 117 

for dot products. 118 
Divergence, 119 

in curvilinear coordinates, 125 
Divergence theorem, 123 
Dot or scalar product, 117, 118 
Double angle formulas, for hyperbolic functions, 27 

for trigonometric functions, 16 
Double integrals, 122 

Duplication formula for gamma functions, 102 
Duplication of cube, 45 

Eccentricity, definition of, 37 

of ellipse, 38 

of hyperbola, 39 

of parabola, 37 
Ellipse. 7, 37, 38 

area of, 7 

eccentricity of, 38 

equation of, 37, 38 

evolute of, 44 

focus of, 38 

perimeter of, 7 

semi-major and-minor axes of, 7, 38 
Ellipses, confocal, 127 
Ellipsoid, equation of, 51 

volume of, 10 
Elliptic cone, 51 

cylinder, 51 

paraboloid, 52 
Elliptic cylindrical coordinates, 127 

Laplacian in, 127 
Elliptic functions, 179-182 [see also Elliptic 
integrals] 

addition formulas for, 180 

derivatives of, 181 

identities involving, 181 

integrals of, 182 

Jacobi's, 180 

periods of, 181 

series expansions for, 181 

special values of, 182 
Elliptic integrals, 179, 180 [see also Elliptic functions] 

amplitude of, 179 

Landen's transformation for, 180 

Legendre's relation for, 182 

of the first kind, 179 

of the second kind, 179 

of the third kind, 179, 180 

table of values for, 254, 255 



Envelope, 44 
Epicycloid, 42 
Equation of line, 34 

general, 35 

in parametric form, 47 

in standard form, 47 

intercept form for, 34 

normal form for, 35 

perpendicular to plane, 48 
Equation of plane, general, 47 

intercept form for, 47 

normal form for, 48 

passing through three points, 47 — 

Error function, 183 

complementary, 183 

table of values of, 257 
Euler numbers, 114, 115 

definition of, 114 

relationship of, to Bernoulli numbers, 115 

series involving, 115 

table of first few, 114 
Euler or Cauchy differential equation, 105 
Euler-Maclaurin summation formula, 109 
Euler's constant, 1 
Euler's identities, 24 
Evolute of an ellipse, 44 
Exact differential equation, 104 
Exponential functions, 23-25, 200 

periodicity of, 24 

relationship of to trigonometric functions, 24 

sample problems involving calculation of, 200 

series for. 111 

table of, 226, 227 
Exponential integral, 183 

table of values for, 261 
Exponents, 23 

F distribution, 189 

95th and 99th percentile values for, 260, 261 
Factorial n, 3 

table of values for, 234 
Factors, 2 
Focus, of conic, 37 

of ellipse, 38 

of hyperbola, 39 

of parabola, 38 
Folium of Descartes, 43 
Fourier series, 131-135 

complex form of, 131 

convergence of, 131 

definition of, 131 

Parseval's identity for, 131 

special, 132-135 
Fourier transforms, 174-178 _ • 

convolution theorem for, 175 

cosine, 176 

definition of, 175 

Parseval's identity for, 175 

sine, 175 

table of, 176-178 
Fourier's integral theorem, 174 
Fresnel sine and cosine integrals, 184 



268 



INDEX 



Frullani's integral, 100 

Frustrum of right circular cone, lateral surface 
area of, 9 
volume of, 9 

Gamma function, 1, 101, 102 

asymptotic expansions for, 102 

definition of, 101, 102 

derivatives of, 102 

duplication formula for, 102 

for negative values, 101 

graph of, 101 

infinite product for, 102, 188 

recursion formula for, 101 

relationship of to beta function, 103 

relationships involving, 102 

special values for, 101 

table of values for, 235 
Gaussian plane, 22 
Gauss' theorem, 123 
Generalized integration by parts, 59 
Generating functions, 137, 139, 146, 149, 161, 153, 

155, 157, 158 
Geometric formulas, 6-10 
Geometric mean, 185 
Geometric series, 107 

arithmetic-, 107 
Gradient, 119 

in curvilinear coordinates, 125 
Green's first and second identities, 124 
Green's theorem, 123 

Half angle formulas, for hyperbolic functions, 27 

for trigonometric functions, 16 
Half rectified sine wave function, 172 
Hankel functions, 138 
Harmonic mean, 185 
Heaviside's unit function, 173 
Hermite polynomials, 151, 162 

addition formulas for, 152 

generating function for, 151 

orthogonal series for, 152 

orthogonality of, 152 

recurrence' formulas for, 151 

Rodrigue's formula for, 151 

special, 151 

special results involving, 152 
Hermite's differential equation, 151 
Higher derivatives, 55 

Leibnitz rule for, 55 
Holder's inequality, 185 

for integrals, 186 
Homogeneous differential equation, 104 

linear second order, 105 
Hyperbola, 37, 39 

asymptotes of, 39 

eccentricity of, 39 

equation of, 37 

focus of, 39 

length of major and minor axes of, 39 
Hyperbolas, confocal, 127 
Hyperbolic functions, 26-31 

addition formulas for, 27 



Hyperbolic functions (cont.) 

definition of, 26 

double angle formulas for, 27 

graphs of, 29 

half angle formulas for, 27 

inverse [see Inverse hyperbolic functions] 

multiple angle formulas for, 27 

of negative arguments, 26 

periodicity of, 31 

powers of, 28 

relationship of to trigonometric functions, 31 

relationships among, 26, 28 

sample problems for calculation of, 200, 201 

series for, 112 

sum, difference and product of, 28 

table of values for, 228-233 
Hyperbolic paraboloid, 52 
Hyperboloid, of one sheet, 51 

of two sheets, 52 
Hypergeometric differential equation, 160 

distribution, 189 
Hypergeometric functions, 160 

miscellaneous properties of, 160 

special cases of, 160 
Hypocycloid, general, 42 

with four cusps, 40 

Imaginary part of a complex number, 21 
Imaginary unit, 21 
Improper integrals, 94 
Indefinite integrals, 57-93 

definition of, 57 

table of, 60-93 

transformation of, 59, 60 
Inequalities, 185, 186 
Infinite products, 102, 188 

series [see Series] 
Initial point of a vector, 116 
Integral calculus, fundamental theorem of, 94 
Integrals, definite [see Definite integrals] 

double, 122 

improper, 94 

indefinite [see Indefinite integrals] 

involving vectors, 121 

line [see Line integrals] 

multiple, 122, 125 
Integration, 67 [see also Integrals] 

constants of, 57 

general rules of, 57-59 
Integration by parts, 57 

generalized, 69 
Intercepts, 34, 47 
Interest, 201, 240-243 
Interpolation, 195 
Interval of convergence, 110 
Inverse hyperbolic functions, 29-31 

definition of, 29 

expressed in terms of logarithmic functions, 29 

graphs of, 30 

principal values for, 29 

relationship of to inverse trigonometric 
functions, 31 

relationships between, 30 



INDEX 



269 



Inverse Laplace transforms, 161 
Inverse trigonometric functions, 17-19 

definition of, 17 

graphs of, 18, 19 

principal values for, 17 

relations between, 18 

relationship of to inverse hyperbolic 
functions, 31 
Involute of a circle, 43 

Jacobian, 125 

Jacobi's elliptic functions, 180 

Ker and Kei functions, 140, 141 
definition of, 140 
differential equation for, 141 
£:raphs of, 141 

Lagrange form of remainder in Taylor series, 110 
Laguerre polynomials, 153, 154 

associated [see Associated Laguerre polynomials] 

generating function for, 153 

orthogonal series for, 164 

orthogonality of, 154 

recurrence formulas for, 163 

Rodrigue's formula for, 153 

special, 153 
Laguerre's associated differential equation, 155 
Laguerre's differential equation, 153 
Landen's transformation, 180 
Laplace transforms, 161-173 

complex inversion formula for, 161 

definition of, 161 

inverse, 161 

table of, 162-173 
Laplacian, 120 

in curvilinear coordinates, 125 
Legendre functions, 146-148 [see also Legendre 
polynomials] 

associated [see Associated Legendre functions] 

of the second kind, 148 
Legendre polynomials, 146, 147 [see also 
Legendre functions] 

generating function for, 146 

orthogonal series of, 147 

orthogonality of, 147 

recurrence formulas for, 147 

Rodrigue's formula for, 146 

special, 146 

special results involving, 147 

table of values for, 252, 253 
Legendre's associated differential equation, 149 

general solution of, 150 
Legendre's differential equation, 106, 146 

general solution of, 148 
Legendre's relation for elliptic integrals, 182 
Leibnitz's rule, for differentiation of integrals, 95 

for higher derivatives of products, 55 
Lemniscate, 40, 44 
Limacon of Pascal, 41, 44 
Line, equation of [see Equation of line] 

integrals [see Line integrals] 

slope of, 34 



Linear first order differential equation, 104 

second order differential equation, 105 
Line integrals, 121, 122 

definition of, 121 

independence of path of, 121, 122 

properties of, 121 
Logarithmic functions, 23-25 [see also Logarithms] 

series for. 111 
Logarithms, 23 [see also Logarithmic functions] 

antilogarithms and [see Antilogarithms] 

base of, 23 

Briggsian, 23 

change of base of, 24 

characteristic of, 194 

common [see Common logarithms] 

mantissa of, 194 

natural, 24 

of complex numbers, 25 

of trigonometric functions, 216-221 

Maclaurin series, 110 

Mantissa, 194 

Mean value theorem, for definite integrals, 94 

generalized, 95 
Minkowski's inequality, 186 

for integrals, 186 
Modified Bessel functions, 138, 139 

differential equation for, 138 

generating function for, 139 

graphs of, 141 

of order half an odd integer, 140 

recurrence formulas for, 139 
Modulus, of a complex number, 22 
Moments of inertia, special, 190, 191 
Multinomial formula, 4 
Multiple angle formulas, for hyperbolic 
functions, 27 

for trigonometric functions, 16 
Multiple integrals, 122 

transformation of, 125 

Napierian logarithms, 24, 196 

tables of, 224, 225 
Napier's rules, 20 
Natural logarithms and antilogarithms, 24, 196 

tables of, 224-227 
Neumann's function, 136 

Nonhomogeneous equation, linear second order, 105 
Normal, outward drawn or positive, 123 

unit, 122 
Normal curve, areas under, 257 

ordinates of, 256 
Normal distribution, 189 
Normal form, equation of line in, 35 

equation of plane in, 48 
Null function, 170 
Null vector, 116 
Numbers, complex [see Complex numbers] 

Oblate spheroidal coordinates, 128 , 

Laplacian in, 128 
Orthogonal curvilinear coordinates, 124-130 

formulas involving, 125 



270 



INDEX 



Orthogonality and orthogonal aeries, 144, 145, 

147, 150, 152, 154, 156, 168, 159 
Ovals of Cassini, 44 

Parabola, 37, 38 

eccentricity of, 37 

equation of, 37, 38 

focus of, 38 

segment of [see Segment of parabola] 
Parabolas, confocal, 126 
Parabolic cylindrical coordinates, 126 

Laplacian in, 126 
Parabolic formula for definite integrals, 95 
Paraboloid elliptic, 62 

hyperbolic, 52 
Paraboloid of revolution, volume of, 10 
Paraboloidal coordinates, 127 

Laplacian in, 127 
Parallel, condition for lines to be, 35 
Parallelepiped, rectangular [see Rectangular 
parallelepiped] 

volume of, 8 
Parallelogram, area of, 5 

perimeter of, 5 
Parallelogram law for vector addition, 116 
Parseval's identity, for Fourier transforms, 175 

for Fourier series, 131 
Partial derivatives, 56 
Partial fraction expansions, 187 
Pascal, limacon of, 41, 44 
Pascal's triangle, 4, 236 
Perpendicular, condition for lines to be, 35 
Plane, equation of [see Equation of plane] 
Plane analytic geometry, formulas from, 34-39 
Plane triangle, area of, 5, 36 

law of cosines for, 19 

law of sines for, 19 

law of tangents for, 19 

perimeter of, 5 

radius of circle circumscribing, 6 

radius of circle inscribed in, 6 

relationships between sides and angles of, 19 
Poisson distribution, 189 
Poisson summation formula, 109 
Polar coordinates, 22, 36 

transformation from rectangular to, 36 
Polar form, expressed as an exponential, 25 

multiplication and division in, 22 

of a complex number, 22, 25 

operations in, 25 
Polygon, regular [see Regular polygon] 
Power, 23 
Power series, 110 

reversion of, 113 
Present value, of an amount, 241 

of an annuity, 243 
Principal branch, 17 
Principal values, for inverse hyperbolic functions, 29 

for inverse trigonometric functions, 17, 18 
Probability distributions, 189 
Products, infinite, 102, 188 

special, 2 
Prolate cycloid, 42 



Prolate spheroidal coordinates, 128 

Laplacian in, 128 
Pulse function, 173 
Pyramid, volume of, 9 

Quadrants, 11 

Quadratic equation, solution of, 32 

Quartic equation, solution of, 33 

Radians, 1, 12, 199, 200 

relationship of to degrees, 12, 199, 200 

table for conversion of, 222 
Random numbers, table of, 262 
Real part of a complex number, 21 
Reciprocals, table of, 238, 239 
Rectangle, area of, 5 

perimeter of, 5 
Rectangular coordinate system, 117 
Rectangular coordinates, transformation of to 

polar coordinates, 36 
Rectangular formula for definite integrals, 95 
Rectangular parallelepiped, volume of, 8 

surface area of, 8 
Rectified sine wave function, 172 

half, 172 
Recurrence or recursion formulas, 101, 137, 139, 

147, 149, 151, 153, 156, 158, 159 
Regular polygon, area of, 6 

circumscribing a circle, 7 

inscribed in a circle, 7 

perimeter of, 6 
Reversion of power series, 113 
Riemann zeta function, 184 
Right circular cone, frustrum of 

[see Frustrum of right circular cone] 

lateral surface area of, 9 

volume of, 9 
Right-handed system, 118 
Rodrigue's formulas, 146, 151, 153 
Roots, of complex numbers, 22, 25 

liable of square and cube, 238, 239 
Rose, three- and four-leaved, 41 
Rotation of coordinates, in a plane, 36 

in space, 49 

Saw tooth wave function, 172 

Scalar or dot product, 117, 118 

Scalars, 116 

Scale factors, 124 

Schwarz inequality [see Cauchy-Schwarz inequality] 

Sector of circle, arc length of, 6 

area of, 6 
Segment of circle, area of, 7 
Segment of parabola, area of, 7 

arc length of, 7 
Separation of variables, 104 
Series, arithmetic, 107 

arithmetic-geometric, 107 

binomial, 2, 110 

Fourier [see Fourier series] 

geometric, 107 

of powers of positive integers, 107, 108 

of reciprocals of powers of positive integers, 
108, 109 



INDEX 



271 



Series, arithmetic (cont.) 
orthogonal [see Orthogonality and orthogonal series] 
power, HO, 113 
Taylor [see Taylor series] 
Simple closed curve, 123 
Simpson's formula for definite integrals, 95 
Sine integral, 183 
Fresnel, 184 
table of values for, 251 
Sines, law of for plane triangle, 19 
law of for spherical triangle, 19 
Slope of line, 34 

Solid analytic geometry, formulas from, 46-52 
Solutions of algebraic equations, 32, 33 
Sphere, equation of, 50 
surface area of, 8 
triangle on [see Spherical triangle] 
volume of, 8 
Spherical cap, surface area of, 9 

volume of, 9 
Spherical coordinates, 50, 126 

Laplacian in, 126 
Spherical triangle, area of, 10 

Napier's rules for right angled, 20 
relationships between sides and angles of, 19, 20 
Spiral of Archimedes, 45 
Square roots, table of, 238, 239 
Square wave function, 172 
Squares, table of, 238, 239 
Step function, 173 
Stirling's asymptotic series, 102 

formula, 102 
Stoke's theorem, 123 
Student's t distribution, 189 
percentile values for, 258 
Summation formula, Euler-Maclaurin, 109 

Poisson, 109 
Sums [see Series] 
Surface integrals, 122 
relation of to double integral, 123 

Tangent vectors to curves, 124 
Tangents, law of for plane triangle, 19 

law of for spherical triangle, 20 
Taylor series, 110-113 

for functions of one variable, 110 

for functions of two variables, 113 
Terminal point of a vector, 116 
Toroidal coordinates, 129 

Laplacian in, 129 
Torus, surface area of, 10 

volume of, 10 
Tractrix, 43 
Transformation, Jacobian of, 125 

of coordinates, 36, 48, 49, 124 

of integrals, 59, 60, 125 
Translation of coordinates, in a plane, 36 

in space, 49 
Trapezoid, area of, 5 

perimeter of, 5 
Trapezoidal formula for definite integrals, 96 
Triangle, plane [see Plane triangie] 

spherical [see Spherical triangle] 



Triangle inequality, 185 
Triangular wave function, 172 
Trigonometric functions, 11-20 

addition formxilas for, 15 

definition of, 11 

double angle formulas for, 16 

exact values of for various angles, 13 

for various quadrants in terms of 
quadrant I, 15 

general formulas involving, 17 

graphs of, 14 

half angle formulas, 16 

inverse [see Inverse trigonometric functions] 

multiple angle formulas for, 16 

of negative angles, 14 

powers of, 16 

relationship of to exponential functions, 24 

relationship of to hyperbolic functions, 31 

relationships among, 12, 15 

sample problems involving, 197-199 

series for, 111 

signs and variations of, 12 

sum, difference and product of, 17 

table of in degrees and minutes, 206-211 

table of in radians, 212-215 

table of logarithms of, 216-221 
Triple integrals, 122 
Trochoid, 42 

Unit function, Heaviside's, 173 
Unit normal to a surface, 122 
Unit vectors, 117 

Vector algebra, laws of, 117 
Vector analysis, formulas from, 116-130 
Vector or cross product, 118 
Vectors, 116 

addition of, 116, 117 

complex numbers as, 22 

components of, 117 

equality of, 117 

fundamental definitions involving, 116, 117 

multiplication of by scalars, 117 

notation for, 116 

null, 116 

parallelogram law for, 116 

sums of, 116, 117 

tangent, 124 

unit, 117 
Volume integrals, 122 

Wallis' product, 188 
Weber's function, 136 
Witch of Agnesi, 43 

X axis, 11 

z intercept, 34 

y axis, 11 

y intercept, 34 

Zero vector, 116 

Zeros of Bessel functions, 250 

Zeta function of Riemann, 184 



60224 



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